Digitized  by  tine  Internet  Arcinive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementsofphysicOOcolericli 


0^xi-eii£.^l-^^>.t.-'^^.-^^     ^-t^c>^3^^^     ^^ 


<d^S-^ 


-»-«—-^ic— »—-•*-' 


THE 

ELEMENTS    OF    F^HV^lC^S 


BY 


S.   E.   COLEMAN,   S.B.,  A.M.  (harvard) 

HEAD  OF  THE   SCIENCE   DEPARTMENT  AND 

TEACHER   OF   PHYSICS    IN  THE   OAKLAND    (CAL.)    HIGH   SCHOOL 

AUTHOR   OF   "a   PHYSICAL   LABORATORY    MANUAL";    HONORARY   MEMBER   OF  THB 

SOUTHERN   CALIFORNIA   ACADEMY  OF  SCIENCES;    PRESIDENT  OF  THB 

PACIFIC  COAST  ASSOCIATION   OF  CHEMISTRY   AND 

PHYSICS  TEACHERS,  1903-1905 


BOSTON,  U.S.A. 

D.   C.   HEATH   &   CO.,   PUBLISHERS 

1906 


Copyright,  1906, 
By  D.  C.  Heath  &  Co. 


r3?^ 


PREFACE 

This  book  is  intended  as  a  text  for  a  first  course  in  physics. 
It  presupposes  a  knowledge  of  elementary  algebra  and  plane 
geometry,  but  requires  no  further  mathematical  training.  While 
complete  in  itself  as  a  text-book,  it  assumes  the  support  of  a 
systematic  course  of  laboratory  exercises  essentially  equivalent 
to  the  author's  Physical  Laboratory  Manual}  The  exercises 
in  this  manual  are  referred  to  by  number  in  the  body  of  the 
text,  in  connection  with  the  topics  which  they  serve  to  illustrate. 
The  text  and  the  manual  together  present  a  single,  coherent 
course  in  elementary  physics. 

The  subject-matter  has  been  selected  with  reference  primarily 
to  its  value  as  a  part  of  a  general  education,  and  includes  an 
unusual  amount  of  information  based  upon  the  facts  of  our  daily 
experience,  introduced  as  illustrations  and  applications  of  phys- 
ical principles.  This  relieves  the  teacher  from  the  necessity  of 
providing  for  and  supervising  a  considerable  amount  of  reference 
reading,  m  order  to  rescue  the  subject  from  the  uninteresting 
barrenness  to  which  it  has  been  reduced  in  many  of  the  more 
recent  text-books. 

The  sequence  of  topics,  especially  in  mechanics,  was  chosen 
with  the  view  of  presenting  the  serious  difficulties  of  the  sub- 
ject gradually  and  in  the  easiest  order.  This  purpose  has  led 
to  a  wide  departure  from  the  conventional  treatment  of  me- 
chanics, which  introduces  a  bewildering  maze  of  abstractions 
at  the  outset  in  connection  with  Newton's  laws  of  motion.  An 
examination  of  the  book  will  easily  disclose  the  author's  reasons 
for  the  order  of  the  major  subdivisions. 

1  Published  by  the  American  Book  Company, 
iii 


iv  Preface 

Classified  problems,  interspersed  at  frequent  intervals,  afford 
opportunity  for  mastery  of  new  ideas  by  their  immediate  use. 
In  addition  to  numerical  examples,  these  problems  present  a 
wide  range  of  familiar  facts  and  phenomena  for  explanation  in 
terms  of  established  laws  and  principles. 

The  Teachers'  Handbook  accompanying  the  text  contains 
(i)  brief  comments  on  the  subject-matter,  including  sugges- 
tions and  references  for  further  experimental  illustration  ;  (2)  a 
list  of  references  to  a  few  more  comprehensive  elementary 
works,  both  general  and  special,  for  supplementary  reading; 
(3)  answers  to  the  problems. 

The  author's  acquaintance  with  the  many  elementary  text- 
books of  physics  already  published  has,  of  course,  been  of 
service  in  the  preparation  of  the  present  work.  This  acquaint- 
ance has  afforded  many  helpful  suggestions,  for  which  no  more 
than  a  general  acknowledgment  is  possible.  The  advanced 
works  and  special  treatises  most  frequently  consulted  were  the 
following:  Atkinson's  translation  of  Ganot's  Physics,  Everett's 
translation  of  Deschanel's  Natural  Philosophy,  Tait's  Properties 
of  Matter,  Madan's  Heat,  S.  P.  Thompson's  Elementary  Lessons 
in  Electricity  and  Magnetism,  and  Huxley's  Lessons  in  Elementary 
Physiology.  Many  facts  and  suggestions  have  been  gathered 
from  these  volumes,  in  addition  to  the  quotations  scattered  here 
and  there  throughout  the  book. 

Special  acknowledgments  are  due  to  Mr.  P.  T.  Tompkins  of 
the  Lowell  High  School,  San  Francisco,  for  reading  the  work 
in  manuscript,  and  to  Dr.  H.  G.  Thomas  of  Oakland  for  his 
critical  examination  of  the  sections  on  the  ear,  the  voice,  and 
the  eye. 

S.  E.  COLEMAN. 

Oakland,  California. 


CONTENTS 


PAGE 

Introduction i 


CHAPTER   I.  — MATTER  AND   FORCE 

I.     The  Three  States  of  Matter 3 

II.     Force  and  Inertia 5 

III.     Measurement lo 


CHAPTER  II.  — THE   MECHANICS   OF   LIQUIDS 

I.     Pressure  produced  in  Liquids  by  their  Weight  .        .        •  15 

II.     Transmission  of  Applied  Pressure  by  Liquids  ...  20 

III.  Buoyancy  of  Liquids 24 

IV.  Specific  Gravity 26 


CHAPTER   IIL  — THE   MECHANICS   OF  GASES 

I.     Atmospheric  Pressure 3^ 

II.     Boyle's  Law 37 

III.     Applications  of  Atmospheric  Pressure       ....  41 

CHAPTER   IV.  — STATICS   OF   SOLIDS 

I.     Concurrent  Forces 49 

II.     Parallel  Forces  ....  .        •        •        •  55 

III.  Moments  of  Force 57 

IV.  Effect  of  Weight  on  the  Equilibrium  of  Bodies         .         .  60. 

CHAPTER   v.  — DYNAMICS 

1.     Motion 68 

II.     Falling  Bodies 74 

V 


vi  Contents 


PAGS 

III.  Projectiles •        .        .      80 

IV.  The  Laws  of  Motion 83 

V.     Curvilinear  Motion 93 

VI.    Universal  Gravitation 98 

VII.    The  Pendulum 104 


CHAPTER  VI.  — ENERGY 

I.     Energy  and  Work Ill 

II.    The  Simple  Machines       .• 123 


CHAPTER  VII.  — MATTER 

I.  The  Structure  of  Matter 137 

II.  Molecular  Motion 141 

III.  Molecular  Forces 147 

IV.  Surface  Tension  and  Capillarity 151 

V.  Properties  Due  to  Molecular  Forces  .        .        .        .156 


CHAPTER  VIII.  — HEAT 

I.     Heat  and  Temperature 161 

II.     The  Transmission  of  Heat 164 

III.  Radiation 167 

IV.  Measurement  of  Temperature  and  Expansion  .        .        .173 
V.    Calorimetry  and  Specific  Heat 182 

VI.     Fusion  and  Solidification 186 

VII.     Vaporization  and  Condensation 192 

VIII.    Mutual  Transformations  of  Heat  and  Other  Forms  of 

Energy       .  ' '.         .  208 


CHAPTER   IX.  — SOUND 

I.     Origin  and  Transmission  of  Sound 214 

II.     Properties  of  Musical  Sounds 227 

III.  Sympathetic  and  Forced  Vibrations :  Resonance      .        .  240 

IV.  The  Ear  and  the  Voice 249 


Contents  vii 


CHAPTER   X.  — LIGHT 

PAGE 

I.     Nature  and  Transmission  of  Light 255 

n.     Reflection  of  Light 268 

HL     Refraction  of  Light 283 

IV.     Atmospheric  Refraction 294 

V.     Lenses 296 

VI.    The  Eye 306 

VII.     Optical  Instruments 315 

VIII.     Dispersion  and  Color 322 

CHAPTER  XL  — MAGNETISM 

I.     Properties  of  Magnets 339 

II.     Magnetization 342 

III.  The  Magnetic  Field 345 

IV.  Terrestrial  Magnetism ,  348 

CHAPTER  XII.  — ELECTRICITY 

I.    The  Voltaic  Cell 353 

11.     The  Electro-magnetic  Field 362 

III.  The  Electric  Bell  and  the  Telegraph         .         .        .         .365 

IV.  Electric  Measurements 369 

V.     Heat,  Light,  and  Power  from  Electric  Currents         .        .  385 

VI.     Induced  Currents 389 

VII.     The  Dynamo  and  the  Motor 396 

VIII.     The  Telephone  and  the  Microphone         ....  404 

IX.    Chemical  Effects  of  Currents 407 

X.     Electrostatics 411 

Appendix  ....  * 427 

Index 431 


or  THF     ^ 

UNIVERSITY 

OF 


ELEMENTS   OF    PHYSICS 


INTRODUCTION 

1.  Physics  and  its  Place  among  the  Sciences.  —  The  knowledge 
of  the  material  universe  is  subdivided  for  convenience  into  several 
branches,  called  the  ?iatural  sciences.  The  biological  sciences 
treat  of  living  things ;  the  physical  sciences  deal  with  inanimate 
matter  in  all  its  forms  and  with  the  changes  and  processes  to  which 
it  is  subject.  The  physical  sciences  are  chemistry,  physics,  as- 
tronomy, geology,  meteorology,  and  mineralogy. 

Chemistry  deals  with  the  changes  in  matter  which  produce  other 
forms  of  matter.  Thus  the  burning  of  wood  is  a  chemical  pro- 
cess, for  it  causes  a  change  of  substance  —  after  the  wood  is 
burned  it  is  no  longer  wood.  Similarly,  the  burning  of  any  sub- 
stance, the  rusting  of  iron,  the  decay  of  animal  or  vegetable 
matter,  and  the  manufacture  of  drugs  and  chemicals  are  chemical 
processes,  and  their  study  falls  to  the  science  of  chemistry. 

Physics  treats  of  changes  and  processes  that  do  not  result  in 
the  formation  of  other  substances.  Its  subdivisions  are  mechanics, 
heat,  sound,  light,  magnetism,  and  electricity. 

Astronomy  is  the  physics  of  the  heavenly  bodies ;  geology,  the 
physics  of  the  earth's  crust ;  meteorology,  the  physics  of  weather 
and  climate  ;  mineralogy,  the  physics  and  chemistry  of  rocks  and 
minerals.  Thus  all  of  the  physical  sciences  are  dependent  upon 
physics,  being,  in  fact,  the  application  of  physics  to  special  fields 
of  study. 

2.  The  Place  of  Physics  in  Education.  —  Physics  deals  largely 
with  familiar  natural  phenomena,^  and  is  therefore  of  special  interest 

^A  phenomenon,  as  the  word  is  used  in  science,  is  "any  action,  motion, 
change,  or  occurrence  of  any  kind,"  however  simple  and  familiar  it  may  be. 

I 


2  Introduction 

and  profit  as  a  part  of  a  general  education.  But  the  value  of  the 
study  of  physics  does  not  consist  alone  in  the  acquisition  of  facts. 
The  pupil  has  already  found  this  to  be  true  of  geometry.  In  this 
subject  the  emphasis  is  laid  not  on  the  fact,  but  on  the  proof  of  it, 
the  main  purpose  being  to  train  the  pupil  in  the  power  and  the 
habit  of  accurate  reasoning.  The  study  of  physics  also  affords  an 
excellent  opportunity  for  valuable  training. 

The  experimental  work  of  the  class  room  and  the  laboratory 
furnishes  the  greater  part  of  the  material  for  this  training ;  but  a 
very  important  part  of  it  is  acquired  through  the  experiences  of 
our  daily  life.  As  the  subject  is  pursued,  this  miscellaneous  body 
of  facts  will  be  passed  in  review,  assorted,  compared,  and  classified, 
and  each  fact  traced  to  its  causes  and  consequences ;  in  a  word, 
the  whole  will  be  organized  into  scientific  knowledge.  The  pupil 
is  receiving  scientific  training  only  in  so  far  as  he  learns  to  do  this 
sort  of  work  for  himself ;  and  it  is  only  this  that  leads  to  scientific 
abitity.  The  mere  memorizing  of  facts  and  explanations  from  the 
text-book  results  in  little  more  than  vaguely  comprehended  and 
therefore  generally  useless  information. 


CHAPTER   I 

MATTER  AND  FORCE 

I.  The  Three  States  of  Matter 

3.  Matter  exists  in  three  conditions  or  states,  called  respectively 
the  solicit  the  liquid^  and  the  gaseous  state.  All  kinds  of  matter  in 
the  same  state  possess  certain  properties  or  characteristics  in  com- 
mon, which  serve  to  distinguish  them  as  a  class  from  bodies  in 
either  of  the  other  states. 

4.  Solids  and  Liquids.  —  Liquids  are  distinguished  from  solids 
by  the  fact  that  they  tend  to  flow,  and  must  therefore  be  contained 
in  vessels.  Solids,  on  the  contrary,  have  a  definite  form,  which 
they  tend  more  or  less  strongly  to  preserve.  Some  solids,  such  as 
stone  and  steel,  offer  great  resistance  to  any  change  of  shape ; 
others,  like  soft  clay  and  putty,  can  readily  be  molded  into  any 
form.  But  even  the  small  amount  of  resistance  offered  by  the 
latter  distinguishes  them  from  liquids. 

5.  Gases.  —  Air  is  the  most  familiar  gas.  Strictly  speaking,  it 
is  a  mixture  of  a  number  of  gases,  principally  nitrogen  and  oxygen  ; 
but  in  physics  it  is  generally  unnecessary  to  consider  the  separate 
constituents.  Although  the  air  is  everywhere  about  us,  we  are  or- 
dinarily unconscious  of  its  existence  unless  it  is  in  motion.  When 
it  is  in  motion,  we  recognize  it  as  a  breeze  or  a  wind,  or,  in  smaller 
quantities,  as  a  current  of  air.  We  commonly  call  a  vessel 
"  empty  "  when  it  is  full  of  air ;  and  seldom  stop  to  think  that, 
while  the  so-called  empty  vessel  is  being  filled  with  any  liquid  or 
solid,  the  air  is  at  the  same  time  being  pushed  out.  It  will  help 
toward  clear  thinking  on  this  point  to  push  an  inverted  tumbler 
into  a  vessel  of  water  (Ex/>.).     The  water  does  not  rise  to  fill  the 

3 


4  Matter  and  Force 

tumbler,  being  prevented  from  doing  so  by  the  confined  air ;  but, 
when  the  tumbler  is  slowly  inclined,  the  air  escapes  in  a  succes- 
sion of  bubbles,  and  water  enters  at  the  same  time  to  take  its 
place. 

The  experiment  shows  that  a  body  of  air  confined  in  any  space 
tends  to  keep  other  bodies  out  of  that  space ;  and  this  is  true  of 
all  gases.  But  it  is  well  known  that,  after  a  bicycle  tire  is  fully 
inflated,  much  air  must  still  be  pumped  in  to  make  it  hard.  Now 
air  can  be  forced  into  the  fully  inflated  tire  only  by  compressing 
the  air  already  in  it  into  a  smaller  space ;  and  experience  teaches 
that  the  compression  of  the  confined  air  can  be  carried  just  as  far 
as  the  strength  of  the  tire  or  of  the  operator  will  permit.  The 
great  compressibility  of  air  can  also  be  shown  by  a  simple  experi- 
ment with  a  bicycle  pump  or  other  compression  pump.  The 
piston  can  be  pushed  in  a  considerable  distance  although  the 
confined  air  is  prevented  from  escaping  by  closing  the  outlet  with 
the  finger.  The  force  required  to  push  the  piston  in  rapidly 
increases,  the  farther  the  piston  is  pushed ;  and,  when  it  is  re- 
leased, it  is  instantly  pushed  back  by  the  expansion  of  the 
compressed  air  (Exp.).  All  gases  are  highly  compressible  and 
expansible,  like  air.  When  any  quantity  of  gas,  however  small, 
is  put  into  an  otherwise  empty  space,  it  expands  so  as  to  fill  the 
space  completely.      {^Experiments  ivith  air  pump.) 

All  liquids  and  most  solids  are  only  very  slightly  compressible. 
In  fact,  any  change  of  volume  with  either  increase  or  decrease  of 
pressure  upon  them  is  so  slight  as  commonly  to  elude  observa- 
tion ;  and,  for  all  practical  purposes,  liquids  are  regarded  as 
incompressible.  Hence  gases  are  distinguished  from  solids  and 
liquids  by  their  great  compressibility  and  expansibility.  Most 
gases  are  colorless  and  invisible,  but  not  all. 

6.  Fluids.  —  Since  the  parts  of  both  liquids  and  gases  move 
over  one  another  freely,  or  flow,  they  are  both  called  fluids.  The 
fluidity  of  gases  is  very  easily  illustrated  with  carbonic  acid  gas ; 
which  will  extinguish  a  lighted  candle  when  poured  upon  it  from 
a  vessel  {Exp.). 


Force  and  Inertia  5 

7.  Summary.  —  Solids  have  both  definite  form  and  volume, 
which  they  tend  to  preserve. 

Liquids  have  definite  volume ;  but  have  no  form  of  their  own, 
since  their  parts  move  readily  over  one  another. 

Gases  have  neither  definite  form  nor  volume.  They  are  highly 
compressible,  and,  with  decrease  of  pressure,  tend  to  expand 
indefinitely. 

II.    Force  and  Inertia 

8.  Force.  —  The  vf 0x6,  force ^  as  used  in  physics,  means  a  push 
or  a  pull.  The  following  are  familiar  examples  of  forces  :  the  pull 
exerted  by  a  horse  upon  a  wagon ;  the  push  or  pull  by  which  a 
door  is  opened  or  closed ;  the  very  brief  but  strong  push  exerted 
by  a  hammer  upon  a  nail  in  driving  it ;  the  continuous  pressure 
of  a  book  lying  upon  a  table ;  the  pressure  of  the  table  upon  the 
book,  by  which  the  book  is  supported ;  the  pressure  of  a  liquid 
against  the  bottom  and  sides  of  the  containing  vessel. 

9.  Inertia.  —  We  learn  from  our  daily  experience  that  a  body 
at  rest  remains  at  rest  unless  some  other  body  exerts  a  force  upon 
it,  or,  in  other  words,  that  a  body  cannot  acquire  motion  without 
the  action  upon  it  of  an  applied  force.  For  example,  a  ball  is  sent 
flying  through  the  air  by  the  push  of  the  hand  in  throwing  it  or  by 
a  blow  with  a  bat ;  a  high  velocity  is  imparted  to  a  rifle  ball  by  the 
pressure  of  the  gases  from  the  powder  exploded  behind  it ;  and  a 
loaded  wagon  is  started  by  the  pull  exerted  by  the  horses  upon  it 
through  the  traces. 

It  is  also  a  matter  of  common  observation  that  moving  bodies 
come  to  rest  more  or  less  slowly  after  the  forces  that  start  them 
cease  to  act.  A  book  slides  over  a  table  when  started  with  a  sud- 
den push,  but  quickly  stops ;  a  ball  can  be  made  to  roll  a  long  dis- 
tance over  a  smooth,  level  surface,  as  a  sidewalk,  but  gradually  loses 
speed  till  it  comes  to  rest ;  and  a  wagon  goes  only  a  short  distance 
after  the  horses  cease  to  pull.  This  behavior  of  moving  bodies  is 
not  due  to  any  tendency  of  the  bodies 'themselves  to  come  to  rest, 
but  is  the  effect  of  opposing  forces  developed  by  the  rubbing  of 


6  Matter  and  Force 

surfaces  as  they  move  over  one  another.  Such  a  force  is  called 
friction.  Friction  acts  as  a  resistance  to  motion,  and  tends  to 
bring  moving  bodies  to  rest.  The  smoother  the  surfaces  are,  the 
less  friction  becomes ;  hence  a  body  slides  farther  on  a  smooth 
surface  than  on  a  rough  one.  For  example,  after  skaters  are  once 
in  rapid  motion,  they  can  go  a  long  distance  without  further  effort, 
the  friction  between  skates  and  smooth  ice  being  very  slight. 
Rolling  friction  is  in  general  much  less  than  sliding  friction ; 
hence  the  use  of  wheels  on  vehicles  of  all  sorts.  Ball  bearings 
reduce  the  friction  still  further  by  substituting  rolling  for  sliding 
friction  at  the  axle. 

Another  hindrance  to  motion  is  the  resistance  of  the  air.  This 
resistance  is  small  upon  a  body  moving  slowly,  but  rapidly 
increases  with  the  velocity.  For  high  velocities,  such  as  those  of 
an  express  train  or  a  rifle  ball,  it  is  very  great.  Bodies  can,  of 
course,  be  stopped  by  other  forces  than  friction. 

The  general  truth  illustrated  by  the  above  examples  may  be 
stated  as  follows :  A  body  remains  at  rest  or,  if  in  motion,  con- 
tinues with  uniform  motion  in  a  straight  line,  unless  it  is  compelled 
to  do  othenvise  by  forces  acting  upon  it  from  without.  This  is  true 
of  all  matter,  solid,  liquid,  and  gaseous;  and  the  property  of 
passiveness  thus  exhibited  is  known  as  the  inertia  of  matter. 

The  inertia  of  water  can  be  shown  by  moving  the  hand  quickly 
to  and  fro  in  a  tub  of  water.  A  strong  push  must  be  exerted  by 
the  hand  to  impart  motion  to  the  water  that  is  driven  before  it. 
The  inertia  of  the  air  is  similarly  shown  by  moving  a  large  fan 
rapidly  to  and  fro,  first  flatwise,  in  the  usual  manner,  then  edge- 
wise. Comparatively  little  effort  is  required  in  the  second  case, 
as  the  fan  cuts  through  the  air,  moving  but  little  of  it.  (Try  these 
experiments.)  Force  is  also  required  to  stop  a  body  of  air  or 
water.  The  resistance  offered  by  a  house  is  sometimes  insufficient 
to  stop  or  turn  aside  the  air  that  strikes  it  during  a  violent  storm, 
and  the  house  is  blown  down.  Similarly,  the  inertia  of  a  stream  of 
water  from  a  fire  hose  or  even  from  a  garden  hose  is  shown  by  its 
power  to  break  or  overturn  obstacles  against  which  it  is  directed. 


Force  and   Inertia  7 

10.  Action  of  Forces  with  and  without  Contact :  Weight.  —  All 
the  forces  considered  in  the  preceding  article  are  exerted  by  direct 
or  indirect  contact  of  the  body  exerting  the  force  and  the  body 
upon  which  the  force  is  exerted.  Thus  a  horse  in  drawing  a 
wagon  pushes  on  the  collar  with  his  shoulders,  and  the  collar 
pulls  on  the  traces,  and  the  traces  pull  on  the  wagon.  Certain 
forces,  however,  act  without  visible  or  material  connection  between 
the  bodies  concerned.  The  forces  exerted  by  a  magnet  are  of  this 
sort.  Pieces  of  iron  move  toward  a  magnet  from  a  greater  or  less 
distance,  depending  upon  the  size  of  the  pieces  and  the  strength 
of  the  magnet  {Exp.).  We  know  from  the  behavior  of  the  iron 
that  it  is  acted  upon  by  a  force  whose  direction  is  toward  the 
magnet,  although  there  is  nothing  whatever  to  show  how  this  force 
is  exerted. 

Similarly,  the  fact  that  a  body  falls  unless  supported  indicates 
that  it  is  acted  upon  by  a  force  whose  direction  is  vertically  down- 
ward. This  force  is  in  some  way  due  to  the  earth ;  hence  we 
think  of  the  earth  as  exerting  a  pull  or  a/traction  by  which  it 
tends  to  draw  bodies  toward  its  center.  The  attraction  exerted 
by  the  earth  upon  any  body  is  called  the  weight  of  the  body.* 

11.  Balanced  and  Unbalanced  Forces.  —  A  force  acting  alone  on 
a  body  always  sets  it  in  motion  or  changes  its  existing  motion.  A 
stone  moving  through  the  air  with  a  moderate  velocity  is  a  good 
illustration ;  for  its  weight  is  practically  the  only  force  acting  on 
it,  the  resistance  of  the  air  being  inappreciable.  The  weight  of 
the  stone  causes  a  continuous  decrease  of  velocity  if  the  stone  is 
rising  vertically,  a  continuous  increase  of  velocity  if  it  is  falling 
vertically,  and  a  continuous  change  of  direction  if  it  is  moving 
obliquely. 

Two  or  more  forces  acting  on  the  same  body  at  the  same  time 
may  neutralize  each  other  in  such  a  way  that  the  body  remains  at 
rest  or,  if  moving,  continues  with  uniform  motion  in  a  straight  line. 
Forces  so  neutralizing  each  other  are  said  to  balance  each  other  or 

1  The  weight  of  a  body  is  very  slightly  less  than  the  earth's  attraction  for^it, 
except  at  the  poles,  as  explained  in  Art.  128. 


8  Matter  and  Force 

to  be  ///  equilibrium^  and  are  called  balanced  forces.  The  follow- 
ing are  illustrations :  A  cart  remains  at  rest  when  two  boys  pull 
equally  on  it  in  opposite  directions.  The  weight  of  a  body  at  rest 
or  in  motion  on  a  level  surface  is  balanced  by 
the  upward  pressure  of  the  surface  ;  the  weight 
of  a  body  suspended  by  a  cord  is  balanced  by 
the  upward  pull  of  the  cord.  In  both  cases  the 
sustaining  force  is  equal  and  opposite  to  the 
weight  of  the  body.  These  examples  illustrate 
the  simplest  case  of  balanced  forces  :  namely, 
that  of  two  equal  forces  acting  in  opposite  di- 
rections along  the  same  line.  An  example  of  three  forces  in  equi- 
librium is  illustrated  in  Fig.  i.  A  body  is  supported  by  two  cords. 
Each  cord  pulls  obliquely  upward  upon  the  body,  and  the  two 
forces  together  balance  the  weight  of  the  body. 

A  single  force  acting  on  a  body  is  always  unbalanced  ;  two  or 
more  forces  acting  together  may  be  either  balanced  or  unbalanced. 
The  study  of  balanced  forces  acting  on  bodies  at  rest  and  of  both 
balanced  and  unbalanced  forces  acting  on  bodies  in  motion  con- 
stitutes the  greater  part  of  the  subject-matter  of  mechanics,  and  is 
continued  in  the  following  chapters. 

12.  The  Mutual  Action  of  Two  Bodies.  —  Whenever  one  body 
exerts  a  force  upon  another,  the  second  body  exerts  at  the  same 
time  an  equal  and  opposite  force  upon  the  first.  This  opposite 
action  of  two  bodies  upon  each  other  is  often  evident  from  the 
effects  produced  upon  the  bodies.  For  example,  when  one  mar- 
ble strikes  another,  the  latter  is  set  in  motion,  and,  at  the  same 
time,  the  motion  of  the  first  marble  is  stopped  or  checked  by  the 
opposite  force  exerted  upon  it  by  the  other  marble.  The  mutual 
action  between  a  ball  and  a  bat  is  a  similar  case.  When  a  bullet 
strikes  a  board,  the  force  that  it  exerts  makes  a  hole  in  the  board  ; 
the  equal  and  opposite  force  exerted  by  the  board  stops  the 
bullet. 

The  equality  of  the  force  exerted  by  each  of  two  bodies  on  the 
other  can  be  illustrated  by  two  hardwood  or  ivory  balls  of  equal 


Force  and  Inertia 


size,  suspended  as  shown  in  Fig.  2.  One  of  the  balls  is  drawn 
aside  and  released.  It  falls,  strikes  the  other  ball,  and  instantly 
stops ;  while  the  other  swings  out  as  far 
(or  very  nearly  as  far)  as  the  first  ball 
would  have  gone  if  its  motion  had  not 
been  hindered.  Since  the  balls  are  ex- 
actly alike  and  the  one  loses  as  much 
motion  as  the  other  gains,  it  follows  that 
the  forces  which  they  exert  upon  each 
other  are  equal  {Exp.). 

The  forces  exerted  between  two  bodies 
at  rest  are  also  equal  and  opposite.  Thus, 
when  the  hand  is  pressed  against  a  wall, 
the  wall  exerts  an  equal  pressure  on  the 
hand.  A  book  lying  on  a  table  exerts  a  downward  pressure  equal 
to  its  weight ;  the  resistance  offered  by  the  table  acts  as  an  equal 
upward  pressure  on  the  book. 

To  distinguish  between  the  action  of  one  body  on  another  and 
the  action  of  the  second  body  on  the  first,  one  is  called  the 
"  action "  and  the  other  the  "  reaction."  Either  may  be  called 
the  action,  and  the  other  will  then  be  referred  to  as  the  reaction ; 
but,  if  one  of  the  bodies  is  at  rest  and  the  other  in  motion  before 
their  mutual  action,  it  is  customary  to  say  that  the  moving  body 
acts  on  the  one  at  rest  and  that  the  latter  reads  on  the  former. 


Fig.  2. 


PROBLEMS 


1.  Describe  and  account  for  the  motion  of  the  occupants  when  a  carriage 
is  started  or  stopped  very  suddenly. 

2.  In  what  direction  is  an  inexperienced  person  likely  to  fall  on  alighting 
from  a  rapidly  moving  car  ?     Why  ? 

3.  What  forces  are  acting  on  a  wagon  when  drawn  at  a  uniform  rate 
on  a  level  road  ?     Are  they  balanced  or  unbalanced  ? 

4.  What  balanced  forces  are  acting  on  a  stone  when  at  rest  on  the  ground? 

5.  A  boy  exerts  a  lifting  force  of  75  lb.  on  a  stone  weighing  200  lb. 
(a)  Is  this  a  balanced  or  an  unbalanced  force  ?  {b)  What  balanced  forces 
are  acting  on  the  stone  ?  • 


lo  Matter  and  Force 

6.  Is  it  the  forces  exerted  by  or  upon  a  body  that  affect  its  state  of  rest  or 
motion  ? 

7.  Account  for  the  "  kick  "  or  recoil  of  a  gun. 

8.  Why  cannot  a  boy  lift  himself  by  standing  in  a  tub  and  pulling  on  the 
handles  ? 

III.    Measurement 

13.  Measurement  and  Units  of  Measurement.  —  Experimental 
work  ill  physics  consists  largely  in  measuring  the  different  kinds  of 
physical  quantities,  as  length,  surface,  volume,  force,  velocity,  mass, 
etc.  A  quantity  of  any  kind  is  measured  by  finding  how  many 
times  it  contains  a  certain  fixed  amount  of  that  kind  of  quantity. 
This  fixed  amount  is  called  a  ///////  and  there  are  various  units  in 
common  use  for  measuring  each  kind  of  quantity.  Thus  there  are 
many  units  of  length,  among  which  are  the  inch,  foot,  meter,  and 
centimeter. 

The  amount  of  any  quantity  is  expressed  by  naming  the  unit  in 
which  it  is  measured,  preceded  by  the  number  of  times  it  con- 
tains this  unit,  —  as  a  volume  of  3.7  cu.  in.  The  name  of  the 
unit  in  which  a  physical  quantity  is  measured  should  not  be 
omitted  either  in  oral  or  written  statement. 

On  account  of  the  great  simplicity  of  the  metric  system  of  meas- 
ures, it  is  almost  exclusively  used  by  scientists ;  and  it  is  the  only 
one  that  we  need  consider  here. 

14.  Extension.  —  Extension  is  that  property  of  matter  by  virtue 
of  which  it  occupies  space,  or  has  length,  width,  and  thickness. 
The  amount  of  space  occupied  by  any  portion  of  matter  is  called 
its  size,  hulk,  or  volume. 

15.  Units  of  Extension.  —  The  fundamental  unit  of  length  in 
the  metric  system  is  the  meter.  It  is  defined  as  the  distance  be- 
tween two  lines  on  a  certain  metallic  rod  preserved  in  the  archives 
of  the  International  Metric  Commission  at  Paris.  It  was  origi- 
nally intended  to  be  one  ten-millionth  of  the  distance  on  the 
earth's  surface  from  the  pole  to  the  equator ;  but  it  is  not  exactly 
this  fraction.    The  meter  is  now  an  arbitrary  standard,  just  as  the 


Measurement  1 1 

yard  is.  Its  advantage  over  the  latter  unit  lies  in  the  fact  that  its 
subdivisions  are  decimal  fractions.    (See  Table  I  of  the  Appendix.) 

A  decimeter  (dm.)  is  a  tenth  of  a  meter  (m.),  a  centimeter  (cm.) 
is  a  hundredth  of  a  meter,  and  a  millimeter  (mm.)  is  a  thousandth 
of  a  meter.  The  centimeter  is  the  customary  unit  of  length  for 
scientific  purposes,  and  is  the  only  one  that  the  pupil  will  ordina- 
rily use  in  the  laboratory.  Thus  a  length  of  3  dm.  5  cm.  7.5  mm. 
is  written  35.75  cm.  It  will  be  useful  to  remember  that  the  meter 
is  equal  to  39.37  inches,  and  the  inch  very  approximately  to  2.5 
cm.  (See  Table  II  of  the  Appendix  for  exact  relative  values  of  the 
English  and  metric  units.) 

The  square  centimeter  (scm.)  and  the  cubic  centimeter  (ccm.) 
are  the  customary  units  of  surface  and  volume  respectively.  Since 
a  square  decimeter  (sdm.)  is  10  cm.  in  length  and  in  width,  it  con- 
tains 100  scm. ;  and  since  a  cubic  decimeter  (cdm.)  is  10  cm.  in 
each  of  its  three  dimensions,  it  contains  1000  ccm.  A  cubic  deci- 
meter, when  used  as  the  unit  of  liquid  measure,  is  called  a  liter. 
It  is  slightly  greater  than  a  quart. 

16.  Weight.  — The  weight  of  a  body  (Art.  10)  is  constant  at  any 
one  place  on  the  earth,  but  decreases  slightly  with  increase  of  alti- 
tude above  the  general  level  of  the  earth  (as  when  a  body  is  carried 
up  a  mountain  or  up  in  a  balloon)  and  also  with  increase  of  depth 
below  the  surface  (as  when  a  body  is  taken  down  into  a  mine).  A 
body  at  the  earth's  center  would  have  no  weight,  being  attracted 
equally  in  all  directions  by  the  earth.  Weight  also  varies  slightly 
with  latitude,  increasing  continuously  upon  any  body  as  it  is  taken 
from  the  equator  toward  either  pole.  The  reasons  for  these 
variations  in  weight  will  be  considered  later. 

17.  Mass.  —  If  any  two  bodies  at  the  same  place  have  equal 
weight,  they  are  said  to  contain  equal  quantities  of  matter^  or  to 
have  equal  mass.  Using  the  term  "  quantity  of  matter  "  in  this 
sense,  the  mass  of  a  body  may  be  defined  as  the  quantity  of  matter 
in  it. 

Although  the  weight  of  a  body  is  affected  by  change  of  lati- 
tude or  altitude,  its  mass  is  not.     A  body  would  contain  identically 


12  Matter  and  Force 

the  same  matter,  and  would  therefore  have  the  same  mass,  if  it 
were  transported  to  some  region  in  space  where  it  would  be  free 
from  the  attraction  of  the  earth  on  any  other  body,  or  to  the  sun, 
where  its  weight  (the  sun's  attraction)  would  be  nearly  twenty-eight 
times  as  great  as  upon  the  earth. 

18.  Units  of  Mass  and  of  Force.  —  The  mass  of  a  cubic  centi- 
meter of  distilled  water  at  the  temperature  of  its  greatest  density 
(4°  Centigrade)  was  originally  taken  as  the  fundamental  unit  of 
mass  in  the  metric  system,  and  is  called  the  gram.  As  in  the  case 
of  the  meter,  the  gram  is  now  defined  with  reference  to  a  standard 
kept  at  Paris ;  but  for  the  purposes  of  elementary  physics  the  above 
definition  is  the  only  one  of  importance.  The  mass  of  a  cubic 
centimeter  of  water  pure  enough  for  domestic  use,  either  cold  or 
tepid,  differs  so  little  from  one  gram  that  the  difference  may  always 
be  disregarded  in  elementary  physics. 

The  earth's  attraction  for  the  unit  of  mass  affords  a  very  con- 
venient unit  of  force.  From  what  has  been  said  concerning  the 
variation  of  weight,  it  is  evident  that  such  a  unit  of  force  differs 
appreciably  at  different  latitudes ;  but  the  variation  is  so  slight  as 
not  to  be  a  matter  of  practical  importance. 

The  unit  of  mass  and  the  corresponding  unit  of  force  have  the 
same  name.  Thus  any  force  equal  to  the  earth's  attraction  for 
a  mass  of  7  g.  would  be  called  a  force  of  7  g.  To  distin- 
guish between  the  units,  they  are  sometimes  called  the  gram  mass 
and  the  gram  iveight  respectively  ;  but  it  is  always  possible  to  de- 
termine from  the  context  whether  mass  or  force  is  intended.  In 
the  English  system  the  pound  mass  and  the  pound  weight  are  the 
fundamental  units  of  mass  and  force  respectively. 

19.  Measurement  of  Mass  (Weighing).  —  The  equal  attraction 
of  the  earth  for  equal  masses  at  the  same  place  is  utilized  in  deter- 
mining the  equality  of  two  masses  by  the  familiar  process  of  weigh- 
ing with  an  equal-arm  balance  (Fig.  3).  The  horizontal  bar  of 
the  balance  is  called  the  beam,  and  either  half  of  it  is  called 
an  arm. 

When  equal  pressures  are  exerted  upon  the  two  pans,  the  beam 


Measurement 


13 


Fig.  3. 


comes  to  rest  in  a  horizontal  position.     Hence,  when  the  beam 
assumes  this  position  under  the  pres- 
sure of  bodies  in  the  two  pans,  these 
bodies  have  equal  mass.     The  reason- 
ing is  as  follows  :  — 

(i)  The  balancing  of  the  beam  in- 
dicates equal  pressures  upon  the  pans. 

(2)  Since  the  bodies  exert  equal 
pressures,  they  have  equal  weight; 
that  is,  the  earth  attracts  them  equally. 

(3)  Since  the  bodies  have  equal 
weight,  they  have  equal  mass. 

The  mass  of  a  body  is  therefore 
found  by  placing  it  in  one  pan  and  balancing  it  with  standard 
masses  in  the  other.  This  process  is  called  weighing,  and  the 
quantity  found  is  commonly  called  the  weight  of  the  body.  The 
standard  masses  are  called  weights. 

20.  Density.  —  The  density  of  a  substance  is  the  mass  of  a  unit 
volume  of  the  substance.  In  the  metric  system  it  is  usually  ex- 
pressed as  the  number  of  grams  in  a  cubic  centimeter  of  it  (g.  per 
ccm.)  ;  in  the  English  system,  as  the  number  of  pounds  in  a  cubic 
foot  of  it  (lb.  per  cu.  ft.). 

The  density  of  any  substance  is  found  by  measuring  the  mass 
and  the  volume  of  any  convenient  portion  of  it,  and  computing 
from  these  measurements  the  mass  of  one  cubic  centimeter. 

Laboraioty  Exercises  J  and  2. 


PROBLEMS 

1.  Would  the  "  weight "  of  a  body,  as  determined  with  an  equal-arm  bal- 
ance^ differ  in  different  latitudes  and  at  different  altitudes?  Give  the  reason 
for  your  answer. 

2.  Is  the  density  of  a  body  affected  by  its  latitude  or  its  altitude  ? 

3.  Is  a  pound  of  iron  heavier  than  a  pound  of  wood  ?  What  is  implied 
in  the  statement  that  "  iron  is  heavier  than  wood  "?  Show  that  the  statement 
"  iron  is  denser  than  wood  "  leaves  nothing  to  be  implied.  The  more  definite 
form  of  statement  is  to  be  preferred. 


1     Cl 


14  Matter  and  Force 


4.  The  volume  of  a  stone  is  630  ccm.;  its  mass  is  1575  g.  Find  its  den- 
sity. 

5.  WTiat  is  the  volume  of  1000  g.  of  mercury  ?  of  icxx)  g.  of  brass  ?  of 
1000  g.  of  aluminum  ?     (See  table  of  densities  in  the  Appendix.) 

6.  What  is  the  mass  of  i  cdm.  of  lead  ?  of  i  cdm.  of  marble  ? 

7.  Find  the  densities  of  water,  quartz,  and  gold  in  pounds,  per  cubic  foot, 
from  the  densities  in  grams  per  cubic  centimeter  given  in  the  table.  (See 
also  the  table  of  equivalents  in  the  Appendix.) 

8.  The  density  of  (luartz  is  how  many  times  that  of  water  in  the  metric 
system  ?  in  the  English  system  ?     How  do  these  answers  compare  ?     "Why  ? 

9.  From  the  known  densities  of  ice  and  water,  show  whether  water  ex- 
pands or  contracts  in  freezing. 


CHAPTER   II 

THE    MECHANICS    OP   LIQUIDS 

I.  Pressure  produced  in  Liquids  by  their  Weight 

21  o  Transmission  of  Pressure.  —  Suppose  a  number  of  books  to 
be  placed  in  a  pile,  one  above  the  other.  Beginning  at  the  top, 
the  first  book  presses  upon  the  second  with  a  force  equal  to  its 
weight.  The  second  book  transmits  this  pressure  to  the  third, 
and  adds  to  it  a  pressure  equal  to  its  own  weight.  Hence  the 
third  book  sustains  a  pressure  equal  to  the  weight  of  the  first  two 
books.  Similarly,  the  third  book  exerts  a  pressure  upon  the  fourth 
equal  to  its  own  weight  plus  that  of  the  books  above  it,  and  so  on 
to  the  last,  the  pressure  of  which  upon  the  table  is  equal  to  the 
combined  weight  of  all.  The  entire  pile  is  supported  by  the  up- 
ward pressure  of  the  table,  which  is  equal  to  the  pressure  of  the 
bottom  book  upon  it  (equal  action  and  reaction) ;  and  any  por- 
tion of  the  pile,  beginning  at  the  top,  is  supported  by  the  upward 
pressure  of  the  book  next  below.  If  we  consider  the  leaves  of 
each  book  separately,  instead  of  the  book  as  a  whole,  we  have  an 
illustration  of  pressure  due  to  weight,  increasing  regularly  and 
continuously  from  top  to  bottom  and  exerted  vertically  up  and 
down,  the  upward  and  downward  pressures  at  any  depth  being 
exactly  equal. 

The  weight  of  a  pile  of  shot  causes  each  shot  to  crowd  in  be- 
tween its  neighbors,  thus  exerting  a  pressure  sideways  as  well  as 
upward  and  downward,  as  is  shown  by  the  tendency  of  the  pile  to 
spread  outward  at  the  bottom.  To  make  the  sides  of  the  pile 
vertical,  supporting  surfaces  must  be  provided  to  sustain  the  lateral 
pressure. 

15 


\ 


u 


1 6  The  Mechanics  of  Liquids 

Similarly,  the  weight  of  a  liquid  causes  lateral  and  oblique  as 
well  as  vertical  pressures  within  it.  These  pressures  are  more 
fully  developed  in  liquids  than  in  a  pile  of  shot,  for  the  particles 
of  a  liquid  are  free  to  move  over  one  another,  while  in  shot  there 
is  considerable  friction.  Hence  shot  remains  in  a  sloping  pile, 
and  liquids  do  not. 

22.  Pressure  in  Liquids  due  to  Weight.  —  The  pressure  produced 
within  a  liquid  by  its  weight  can  be  studied  experimentally  with 
various  forms  of  apparatus.  Cilass  tubes  of  small 
diameter  and  equal  length  (about  60  cm.),  closed 
air-tight  at  the  top  and  shaped  at  the  lower  end  as 
shown  in  Fig.  4,  serve  very  well  for  the  purpose 
of  illustration,  but  are  not  adapted  to  exact  meas- 
urement. When  one  of  the  tubes  is  lowered  into 
a  tall  glass  jar  filled  with  water,  the  water  enters 
its  lower  end  a  short  distance.  This  is  due  to 
the  compression  of  the  confined  air  by  the  pressure 
Fig.  4.  that  the  water  exerts  upon  it.     The  distance  to 

which  the  water  enters  the  tube  increases  as  the 
tube  is  lowered,  showing  that  the  pressure  increases  with  the 
depth  {Exp.).  This  is  due  to  the  fact  that  the  pressure  at  any 
depth  is  caused  by  the  weight  of  the  water  above  that  level. 

When  the  different  tubes  are  inserted  to  the  same  depth,  the 
water  enters  an  equal  distance  in  all,  showing  that  the  pressure 
of  the  water  at  a  given  depth  is  the  same  in  the  various  directions 
in  which  it  enters  the  tubes  {Exp.),  Since  each  particle  of  water 
is  free  to  move  in  any  direction,  it  would  not  remain  at  rest  if  it 
were  not  pressed  upon  equally  from  all  sides. 
Laboratory  Exercise  j". 

23.   Laws  of  Liquid  Pressure.  —  The  facts  concerning  pressure  in 
liquids  at  rest  are  found  by  accurate  measurement  to  be  as  follows  :  — 

I.  The  pressure  at  any  point  in  a  liquid  at  rest  is  equal  in  all 
directions. 

II.  The  pressure  of  a  liquid  at  rest  is  perpendicular  to  any  surface 
upon  which  it  acts. 


Pressure  in   Liquids  17 

III.  At  any  point  ill  a  liquid^  the  pressure  due  to  its  weight  is 
proportional  to  the  depth  of  the  point  below  the  free  surf  ace  of  the 
liquid. 

IV.  The  pressure  is  the  same  at  all  points  in  the  same  horizontal 
plane. 

V.  At  the  sattie  depth  in  different  liquids,  the  pressure  due  to 
weight  is  proportional  to  the  density  of  the  liquid. 

These  statements  are  known  as  the  laws  ^  of  liquid  pressure. 
The  first  two  hold  for  all  pressures  in  liquids  at  rest,  whether  due 
to  their  weight  or  to  applied  pressure,  i.e.  to  pressure  exerted 
upon  them  in  closed  vessels.  As  stated  in  the  preceding  article, 
the  first  law  is  a  consequence  of  the  freedom  of  movement  of  the 
particles  of  a  liquid.  This  is  true  of  the  second  law ;  for,  if  a 
liquid  pressed  obliquely  against  a  surface,  it  would  move  along  the 
surface  instead  of  remaining  at  rest.  The  reasons  for  the  other 
laws  are  considered  in  the  following  articles. 

24.  Intensity  of  Pressure.  —  The  pressure  at  any  point  in  a  liquicj 
is  defined  as  the  pressure  that  the  liquid  would  exert  upon  a  hori- 
zontal surface  of  //////  area  at  that  depth  in  the  liquid.  The  pres- 
sure upon  each  unit  area  of  a  surface  is  sometimes  called  the 
intensity  of  pressure  or  the  rate  of 
pressure  to  distinguish  it  from  the 
/^/^/ pressure  on  the  surface.  Pres- 
sure (/.<?.  intensity  of  pressure)  is 
expressed  in  grams  per  square  centi- 
meter, pounds  per  square  foot,  pounds 
per  square  inch,  and  other  units  con- 
sidered later.  ^,,  ,  ^ 

The  rule  for  computing  the  pressure 
at  any  depth  in  a  liquid  is  derived  as  follows  :•  Let  MN  (Fig.  5 ) 

1  Experience  teaches  that,  in  nature,  whenever  the  same  conditions  are  repeated 
the  same  resuhs  follow ;  or,  in  other  words,  the  same  causes  always  produce  the 
same  effects.  Without  this  uniformity  in  nature  science  would  be  impossible.  The 
uniform  behavior  of  matter  under  given  conditions  is  called  a  natural  law.  A  brief 
descriptive  statement  of  such  uniform  behavior  in  any  given  instance  is  also  called 
a  law. 


1 8  The  Mechanics  of  Liquids 

be  a  horizontal  surface  upon  which  a  liquid  rests,  and  a  any  square 
centimeter  of  it.  The  rectangular  figure  ac  represents  the  portion 
of  the  liquid  lying  vertically  above  a.  We  may  think  of  this  por- 
tion as  being  separated  from  the  remainder  by  imaginary  bounding 
surfaces,  or  as  having  become  solid  (without  change  of  density). 
The  surrounding  liquid  presses  perpendicularly  against  the  sides 
of  this  column,  as  indicated  by  the  arrows.  The  pressures  upon 
opposite  sides  are  equal  and  balance  each  other,  but  they  do  not 
help  to  sustain  the  weight  of  the  column.  The  pressure  that  the 
column  exerts  on  a  is  therefore  equal  to  its  weight.  Let  //  denote 
the  height  of  the  column  in  centimeters,  and  d  the  density  of  the 
liquid  in  grams  per  cubic  centimeter;  then  the  volume  of  the 
column  is  h  ccm.  and  its  weight  hd  g.     Hence  the  rule  :  — 

At  any  point  in  a  liquid^  the  pressure  due  to  its  weight  is  equal 
to  the  product  of  the  depth  {of  that  point  below  the  free  surface)  and 
the  density  of  the  liquid. 

25.  Total  Pressure  upon  a  Surface.  —  Horizontal  Surfaces.  — 
Since  the  pressure  of  a  liquid  is  uniform  over  a  horizontal  surface, 
the  total  pressure  on  such  a  surface  is  equal  to  the  product  of  the 
intensity  of  pressure  and  the  area. 

Oblique  and  Vertical  Surfaces,  —  The  pressure  upon  a  vertical 
or  an  oblique  plane  surface  {MN, 
Fig.  6)  increases  regularly  from  the 
upper  to  the  lower  side.  To  find  the 
total  pressure  upon  such  a  surface, 
the  average  pressure  (per  scm.)  upon 
it  is  multiplied  by  the  area.  The 
average  pressure  is  equal  to  the  actual 
pressure  at  the  center  of  the  surface  ; 
and  the  latter  is  ^he  same  as  the 
pressure  upon  a  horizontal  area  of  i  scm.  at  that  depth,  since 
the  pressure  at  any  point  is  equal  in  all  directions. 

For  example,  the  total  pressure  upon  an  oblique  surface  of  4 
cm.  by  12  cm.,  the  center  of  which  is  at  a  depth  of  10  cm.  in 
alcohol  (density  =  .82  g.  per  scm.)  is  4  x  12  x  10  x  .82  g. 


Pressure  in   Liquids 


19 


Fig.  7. 


The  average  pressure  on  the  side  of  a  cylindrical  or  rectangular 
vessel  containing  a  liquid  is  the  actual  pressure  at  half  the  depth 
of  the  liquid. 

26.  Pressure  Independent  of  Shape  of  Vessel.  —  One  important 
consequence  of  the  laws  of  liquid  pressure  is  that  the  pressure  at 
any  point  in  a  liquid  is  independent 
of  the  shape  of  the  containing  vessel 
{Exp.) .  Suppose  that  vessels  A,  B, 
Cy  and  D  (Fig.  7)  each  have  a  bot- 
tom 10  X  10  cm.  and  a  height  of  30 
cm.,  and  are  filled  with  water.  The 
lower  portion  of  B  is  10  cm.  high, 
and  is  continued  20  cm.  higher  by 
a  tube.  The  pressure  will  be  uniform  over  the  bottoms  of  the 
four  vessels  and  will  be  equal  upon  all ;  namely,  30  g.  per  scm., 
or  a  total  of  3000  g.  upon  each.  This  is  evidently  the  case  for  A^ 
since  3000  g.  is  the  weight  of  the  water  it  contains. 

The  pressure  at  the  bottom  of  the  tube  in  B  is  20  g.  per  scm. 
This  causes  an  equal  upward  pressure  (per  scm.)  against  the  top 
of  the  lower  portion  of  the  vessel,  as  indicated  by  the  arrows  in 
the  figure.  This  upward  pressure  of  the  water  is  sustained  by  an 
equal  downward  pressure  of  the  top  against  the  water.  This 
downward  pressure  of  the  top  takes  the  place  of  an  equal  down- 
ward pressure  at  the  same  level  in  A^  due  to  the  weight  of  the 
water  above  that  level.  Hence  the  pressures  in  the  two  vessels  at 
equal  depths  below  this  level  are  equal. 

In  C  the  weight  of  all  the  liquid  not  lying  vertically  above  the 
bottom  is  sustained  by  the  sides  of  the  vessel,  which  exert  an 
oblique  upward  pressure.  The  discussion  of  pressures  for  D  is 
similar  to  that  for  B.     (State  it.) 

27.  Equilibrium  of  a  Liquid  in  Communicating  Vessels.  —  If  the 
vessels  represented  in  Fig.  7  were  connected  with  one  another  by 
tubes  as  indicated  by  the  dotted  lines  in  the  figure,  there  would 
be  no  flow  to  or  from  any  one  of  them  ;  for  the  pressure  would  be 
the  same  at  lx)th  ends  of  each  of  the  tubes  {Exp.). 


20 


The  Mechanics  of  Liquids 


Illustrations  of  this  fact  are  common.  The  liquid  stands  at 
the  same  level  in  the  spout  of  a  teapot  as  in  the  vessel  itself. 
The  pressure  at  the  bottom  of  the  spout  must  be  the  same  on  both 
sides,  otherwise  the  liquid  would  not  remain  at  rest.  Similarly,  in 
a  pipe  leading  from  a  reservoir  the  water  will  rise  to  the  level  of 
the  water  in  the  reservoir  and  no  higher.     This  familiar  behavior 

of  water  is  commonly  expressed 
by  saying  that  "  water  seeks  its 
own  level."  This  is,  of  course, 
merely  a  statement  of  the  fact, 
not  an  explanation  of  it. 

The  flow  of  artesian  wells  is 
explained  by  Fig.  8.  If  a  por- 
ous stratum  of  sand  or  gravel, 
/I,  lying  between  two  imper- 
vious strata  and  dipping  under 
a  lower  flat  country,  becomes  filled  with  water  above  the  level  of 
the  groimd  where  a  well  is  bored,  an  artesian  or  flowing  well 
will  result. 


FiC.  8. 


II.  Transmission  of  Applied  Pressure  by  Liquids 

28.  Pascal's  Law. — Pressure  exerted  upon  any  part  of  an 
inclosed  liquid  is  transmitted  undiminished  in  all  directions^  and 
acts  with  equal  force  upon  all  equal  surfaces  ^  and  at  right  angles  to 
them.  This  is  known  as  Pascal's  law,  in  honor  of  the  French 
mathematician  and  physicist,  Blaise  Pascal  (1623- 
1662),  by  whom  it  was  discovered. 

For  example,  if  a  stopper  having  an  area  of 
2  scm.  at  the  end  be  pushed  into  a  botde  full  of 
water  until  it  presses  with  a  force  of  6  kg.  upon 
the  water,  this  pressure  of  3  kg.  per  scm.  will  be 
added  to  the  pressure  due  to  the  weight  of  the 
liqtiid  at  every  point  within  it,  and  will  be  trans- 
mitted as  an  added  pressure  of  3  kg.  upon  each  fig.  9. 


Transmission  of  Pressure  21 

square  centimeter  of  surface  of  the  bottle.     With  the  apparatus 

shown  in  Fig.  9,  a  strong  bottle  can  be  easily  burst  by  pushing 

the  rod  a  short  distance  through  the  stopper  after  the  bottle  has 

been  filled  and  closed  tightly.     In  this  way  it  would  require  only 

a  moderate  effort  to  cause  a  total  pressure  of  a  ton  on  the  interior 

surface  of  the  bottle. 

Pascal's  law  is  indirectly  contained  in  the  laws  previously  stated. 

Thus,  in  vessel  A  of  Fig.  7  it  is  clear 

that  the  pressure  at  every  point  below 

the  level  ae  (see  discussion  in  Art. 

26)  is  20  g.  per  scm.  greater  than  it 

would  be  if  the  water  were  removed 

from  above  that  level.     That  is,  the 

pressure  exerted  by  the  portion  of 

f,  ,  .  .      J  Fig.  10. 

the  water   above  ae  is   transmitted 

throughout  the  portion  below  that  level  in  accordance  with  Pascal's 

law. 

29.  The  Hydrostatic  Press.  —  It  will  be  seen  from  the  fore- 
going that  water  is  a  very  effective  instrument  for  transmitting  and 
increasing  force.  This  is  utilized  in  the  hydrostatic  (or  hydraulic) 
press,  a  machine  invented  over  one  hundred  years  ago.  The 
principle  of  the  hydrostatic  press  is  shown  in  Fig.  10.  Two 
cylinders  of  different  diameters  are  connected  by  a  tube  and  filled 
with  water.  They  are  fitted  with  tight  pistons,/  and  /*,  resting 
upon  the  water.  Let  the  area  of  the  larger  piston  be  fifty  times 
that  of  the  smaller.  A  weight  of  i  kg.  placed  on  the  smaller 
piston  will  cause  the  water  to  exert  a  pressure  of  i  kg.  upon  every 
2irt2i  equal  to  that  of  the  smaller  piston.  Hence  the  total  upward 
pressure  upon  the  larger  piston  will  be  50  kg.,  and  will  sustain  a 
weight  of  50  kg. 

One  form  of  hydrostatic  press  is  shown  in  Fig.  11.  ^  is  a  very 
strong  metal  cylinder.  In  it  there  is  a  cast  iron  piston,  /*,  work- 
ing water-tight  in  the  collar  of  the  cylinder.  The  top  of  the 
piston  carries  an  iron  plate  on  which  the  substance  to  be  pressed 
is  placed.    The  fixed  upper  plate,  Q,  is  supported  upon  four  strong 


22 


The  Mechanics  of  Liquids 


columns.  Water  is  pumped  into  the  cylinder  through  the  pipe,  K^ 
by  means  of  a  force  pump,  A  (see  Art.  52).  When  the  piston  of 
the  pump,  /,  is  forced  down,  by  the  downward   stroke  of  the 


Fig.  II. 


handle  My  it  exerts  a  pressure  upon  the  water ;  and  this  pressure 
is  transmitted  through  the  water  in  the  pipe  and  in  the  cylinder 
to  the  lower  end  of  P,  forcing  it  upward.  The  force  exerted  on 
the  piston  of  the  press  is  to  the  force  exerted  by  the  piston  of  the 
pump  as  the  area  of  the  end  of  the  first  piston  is  to  the  area  of 
the  end  of  the  second  piston. 

The  hydrostatic  press  is  used  for  many  purposes  where  great 
pressures  are  required.  It  is  used  in  compressing  loose  cotton 
into  bales,  in  bending  iron  plates,  in  forcing  car  wheels  on  to  the 
axles,  in  lifting  heavy  weights,  etc.  The  more  powerful  presses 
have  pumps  operated  by  engines  and  are  capable  of  exerting 
pressures  of  several  hundred  tons. 


Transmission  of  Pressure 


23 


PROBLEMS 


1.  The  free  surface  of  a  liquid  at  rest  is  level.  What  is  the  difference 
between  a  level  surface  and  a  horizontal  plane  surface  ?  Why  is  the  dis- 
tinction commonly  disregarded  ? 

2.  Why  is  the  surface  of  a  liquid  at  rest  level  ? 

3.  What  is  the  pressure  (a)  at  a  depth  of  20  cm.  in  water  ?  (d)  at  a  depth 
of  60  cm.  in  mercury  (see  table  of  densities  in  the  Appendix)  ?  (f)  at  a  depth 
of  50  cm.  in  alcohol  ? 

4.  What  is  the  pressure  in  pounds  per  square  foot  (a)  at  a  depth  of  20 
ft.  in  water  ?  (6)  at  a  depth  of  3  miles  in  the  ocean?  (Take  62.4  lb.  per 
cu.  ft.  as  the  density  of  pure  water  in  all  problems.  The  density  of  sea 
water  is  1.026  times  this.) 

5.  What  departure  from  the  proportionality  of  depth  and  pressure  would 
result  if  liquids  were  appreciably  compressible  ? 

6.  A  rectangular  vessel  50  cm.  long,  20  cm.  high,  and  35  cm.  wide  is 
filled  with  a  liquid  whose  density  is  1.5  g.  per  ccm.  Find  the  total  pressure 
(a)  upon  the  bottom,  (6)  upon  a  side. 

7.  Find  the  total  pressure  in  pounds  against  the  side  of  a  cylindrical  tank 
15  ft.  in  diameter  and  12  ft.  high,  when  filled  with  water. 

8.  A  cask  2  ft.  high  and  18  in.  in  average  diameter  is  fitted  with  an  iron 
pipe  40  ft.  long,  as  shown  in  Fig.  1 2.  The  cross- 
section  of  the  pipe  is  .25  sq.  in.  Cask  and  pipe 
are  filled  with  water,  (a)  Compute  the  total 
pressure  against  the  side  of  the  cask,  (d)  What 
weight  of  water  does  the  tube  hold  ?  (c)  How 
would  the  pressure  within  the  cask  be  affected 
by  using  a  pipe  having  a  cross-section  of  2.5  sq. 
in.? 

Note.  —  Pascal  succeeded  in  bursting  a  very 
strong  cask  in  the  manner  here  indicated,  using 
a  slender  tube  40  ft.  high. 

9.  If  the  diameter  of  the  piston  in  the  pump 
of  an  hydraulic  press  is  2  cm.  and  that  of  the 
piston  of  the  press  is  30  cm.,  what  will  be  the 
total  pressure  upon  the  latter  due  to  a  force  of 
50  kg.  upon  the  former  ? 

10.  What  is  the  total  pressure  against  a  vertical  dam  100  ft.  long,  against 
which  the  water  stands  to  a  depth  of  12  ft.  ? 

11.  A  cylindrical  vessel  20  cm.  in  diameter  is  filled  to  a  depth  of  15  cm. 
with  oil  having  a  density  of  .85  g.  per  ccm.     A  piston  fitting  the  vessel  tightly 


Fig.  12. 


24  The   Mechanics  of  Liquids 

is  pushed  down  upon  the  oil  with  a  force  of  lOO  kg.  What  is  the  total  pres- 
sure upon  the  side  of  the  vessel  due  to  the  weight  of  the  liquid  and  the  pres- 
sure of  the  piston  ? 

III.    Buoyancy  of  Liquids 

30.  Buoyancy.  —  It  is  a  familiar  fact  that  a  liquid  either  wholly 
or  partly  supports  any  solid  placed  in  it.  The  supporting  force 
exerted  by  a  liquid  upon  a  body  either  wholly  or  partly  immersed 
in  it  is  called  buoyant  force  or  buoyancy.  If  the  buoyant  force 
upon  a  body  is  equal  to  its  weight,  the  body  will  float ;  if  less,  it 
will  sink,  but  its  apparent  weight  in  the  liquid  (measured  by  the 
force  required  to  support  it  while  immersed)  will  be  less  than  in 
air. 

Laboratory  Exercise  6. 

31.  The  Principle  of  Archimedes.  — The  law  of  buoyancy  was 
discovered  by  Archimedes,  a  celebrated  Greek  mathematician  of 
the  third  century  b.c.  It  is  known  as  iht  principle  of  Archimedes j 
and  is  as  follows  :  A  body  either  wholly  or  partly  immersed  in  a 
liquid  is  buoyed  up  by  a  force  equal  to  the  weight  of  the  liquid  dis- 
placed by  it. 

The  proof  of  this  law  in  the  case  of  a  cylindrical  or  rectangular 
body,  wholly  immersed  and  vertical,  is  as  follows :  Let  a  scm.  be 
the  area  of  the  base  and  //  cm.  the  height  of 
the  body  (Fig.  13),  and  let  its  top  be  at  a 
depth  of //cm.  below  the  surface  of  a  liquid 
whose  density  is  d  g.  per  ccm.  The  pres- 
sures upon  opposite  sides  balance  each 
other ;  but  they  do  not  tend  to  support  the 
body.  The  pressure  upon  the  top  is  Hd  %. 
per  scm.,  and  the  total  pressure  aHd  g.,  or 
"  '^*  the  weight  of  the  column   of  liquid   lying 

vertically  above  the  solid.  Since  the  bottom  is  {H-\-h)  cm.  below 
the  surface,  the  pressure  upon  it  is  {H-\-h)d  g.  per  scm. ;  and  the 
total  pressure  a  (//-\-h)  dg.^  or  the  weight  of  a  column  of  the  liquid 
whose  base  is  a  and  whose  height  is  H-^h.     The  buoyant  force 


Buoyancy  of  Liquids 


25 


is  the  difference  between  these  upward  and  downward  forces,  that 
is,  a  (^H-yh)  —aHd  or  ahd  g. ;  which  is  equal  to  the  weight  of  the 
displaced  liquid. 

If  the  body  is  sunk  to  a  greater  depth,  the  pressures  upon  its 
top  and  bottom  increase  by  equal  amounts,  hence  the  buoyant 
force  remains  constant. 

The  following  proof  of  the  law  holds  for  a  body  of  any  shape. 
Let  A  (Fig.  14)  represent  any  portion  of  a  liquid, 
having  any  shape  whatever.  Since  this  portion 
remains  at  rest,  its  weight  is  evidently  balanced 
by  an  equal  buoyant  force,  due  to  the  pressure  of 
the  surrounding  liquid.  These  pressures  would 
remain  the  same  if  A  were  replaced  by  any  solid 
having  the  same  size  and  shape;  hence  the 
buoyant  force  upon  the  solid  would  be  the  same 
as  it  was  upon  the  displaced  liquid  ;  i.e,  it  would 
be  equal  to  the  weight  of  the  liquid  displaced  by  the  solid. 

32.  Buoyancy  upon  Floating  Bodies.  —  If  the  density  (or  aver- 
age density)  of  a  body  is  greater  than  that  of  the  liquid  in  which 
it  is  immersed,  its  weight  will  exceed  the  buoyant  force  and  it  will 
sink  unless  otherwise  supported.  If  the  density  of  the  body  is 
less  than  that  of  the  liquid,  its  weight  will  be  less  than  the  buoyant 
force  upon  it  when  completely  immersed  ;  hence,  if  not  held  down, 
it  will  be  pushed  to  the  surface  and  will  float  partly  immersed. 

The  buoyant  force  on  a  floating  body  is  equal  to  the  weight  of 
the  liquid  displaced  by  the  immersed  portion  of  the  body ;  it  is 
also  equal  to  the  weight  of  the  body,  since  the  two  forces  are  in 
equilibrium.  Hence  a  floating  body  displaces  its  own  weight  of 
the  liquid  in  which  it  floats. 


Fig.  14. 


PROBLEMS 

1.  Will  a  ship  sink  to  a  greater  or  less  depth  on  sailing  from  fresh  into 
salt  water  ? 

2.  From  the  fact  that  ice  floats  in  water,  what  may  b%  inferred  concerning 
the  change  of  volume  of  water  in  freezing  ? 


26  The  Mechanics  of  Liquids 

3.  From  the  table  of  densities  in  the  Appendix,  make  a  list  of  metals  that 
will  float  in  mercury,  and  a  list  of  those  that  will  sink  in  it. 

4.  Since  steel  is  many  times  denser  than  water,  how  can  a  ship  made  of 
steel  float  ? 

5.  Experiments  have  shown  the  compressibility  of  sea  water  to  be  suohf- 
that,  at  a  depth  of  a  mile,  its  density  is  j^^f  greater  than  at  the  surface, 
(a)  Would  a  body  that  sinks  in  sea  water  at  the  surface  be  likely  to  sink 
to  the  bottom   of  the   ocean,  however   deep  ?     (^)  Under  what   condition 
would  it  not  do  so  ? 

6.  The  density  of  sea  water  is  1.026  g.  per  ccm.  and  that  of  ice  is  .917  g. 
per  com.     What  portion  of  an  icel)erg  is  immersed  ? 

Suggestion.  —  The  immersed  portion  of  the  iceberg  is  equal  in  volume 
to  the  displaced  water,  and  the  weight  of  the  displaced  water  is  equal  to  that 
of  the  iceberg. 

7.  Account  for  the  relative  position  of  two  or  more  liquids  that  will  not 
mix  (as  oil,  water,  and  mercury)  when  placed  in  the  same  vessel  (^Exp."), 

IV.    Specific  Gravity 

33.  Specific  Gravity  Defined.  —  The  ratio  of  the  density  of  a 
substance  to  the  density  of  distilled  water  at  4°  Centigrade  is 
called  the  specific  gravity  (or  specific  density)  of  the  substance. 
Thus,  if  the  density  of  a  stone  is  150  lb.  per  cu.  ft.,  its  specific 
gravity  is  150-5-62.4,  or  2.44,  62.4  lb.  per  cu.  ft.  being  the  density 
of  pure  water  at  4°  C.  Hence  to  say  that  the  specific  gravity  of 
the  stone  is  2.44  means  that  the  stone  is  2.44  times  as  dense  as 
water. 

Since  the  density  of  water  is  i  g.  per  ccm.,  the  density  of  the 
stone  in  the  above  problem  is  2.44  X  i  g.,  or  2.44  g.  per  ccm. 
Thus,  in  the  metric  system,  specific  gravity  and  density  are  numer- 
ically equal.^  This  is  also  evident  from  the  definition,  since,  in 
the  metric  system,  the  numerical  value  of  the  second  term  of  the 
ratio  is  one. 

Except  where  great  accuracy  is  required  in  scientific  work,  it 
is  unimportant  to  take  account  of  the  very  slight  difference  be- 

1  The  equality  of  density  and  specific  gravity  in  the  metric  system  is  only  numer- 
ical ;  the  quantities  are  different  in  kind,  as  is  clearly  evident  when  English  units 
are  used.     A  ratio  is  always  an  abstract  number. 


specific  Gravity 


27 


tween  the  density  of  fresh  water  at  ordinary  temperatures  and  the 
density  of  distilled  water  at  4°  C. ;  hence  the  specific  gravity  of  a 
substance  may  be  defined  as  the  ratio  of  its  density  to  the  density  of 
water.  For  the  purpose  of  laboratory  work,  it  is  useful  to  define 
the  specific  gravity  of  a  substance  as  the  ratio  of  the  mass  (or 
weight)  of  any  volume  of  it  to  the  mass  (or  weight)  of  an  eqtial 
volume  of  water.  (Show  that  this  definition  is  consistent  with  the 
first ;  i.e.  that  they  are  not  independent  and  possibly  contradictory 
definitions.) 

34.  Methods  of  Finding  Specific  Gravity.  —  If  the  density  of  a 
substance  in  pounds  per  cubic  foot  is  known,  its  specific  gravity  is 
found  by  dividing  this  number  by  62.4.  If  the  density  is  known 
in  the  metric  system,  its  specific  gravity  is  known  without  further 
computation. 

There  are  several  simple  experimental  methods  for  finding  the 
specific  gravity  of  a  substance  without  the  necessity  of  determining 
its  density.     The  ones  most  frequently  used  are  presented  in  the"^ 
following  articles.     They  depend,  for  the  most  part,  upon  the  prin-  • 
ciple  of  Archimedes,  and  hold  without  modification  for  either  Eng- 
lish or  metric  units  of  mass  and  volume. 

35.  Specific  Gravity  of  Solids.  —  Solids  Denser  than  Water.  — 
The  solid  is  weighed,  then  sus- 
pended by  a  thread  from  a  pan  of 
the  balance,  and  again  weighed 
while  hanging  wholly  immersed  in 
water  (Fig.  15).  The  difference 
between  these  weights  measures  the 
buoyant  force  upon  the  body,  and 
is,  therefore,  the  weight  of  an  equal 
volume  of  water.  Hence  the  ex- 
periment gives  the  masses  of  equal 
volumes  of  the  solid  and  of  water. 
The  specific  gravity  of  the  solid 
is  the  ratio  of  its  mass  to  the  mass  of  the  displaced  water. 

Solids  Less  Dense  than  Water,  —  In  this  case  a  denser  body, 


Fig.  15. 


28  The  Mechanics  of  Liquids 

called  a  sinker^  is  attached  to  the  solid  to  keep  it  wholly  immersed 
when  weighed  in  water.  The  relations  involved  will  be  understood 
from  the  following  example  :  — 

\Veight  of  lK)dy  in  air  =  40  g. 

Weight  of  sinker  in  water  =  35  g- 

Weight  of  body  and  sinker  together  in  water  =  15  g. 

It  will  be  seen  that  the  buoyant  force  upon  the  immersed  body 
sustains  its  entire  weight  and  also  20  g.  of  the  weight  of  the 
sinker  (35  —  15).  That  is,  the  buoyant  force  upon  the  body  ex- 
ceeds its  weight  by  20  g.  Its  value  is,  therefore,  40  +  20  g.,  or 
60  g. ;  and  this,  by  Archimedes'  principle,  is  the  weight  of  water 
equal  in  volume  to  the  body.  Hence  the  computations  are  as 
follows :  — 

Amount  by  which  buoyancy  upon  the  body 

exceeds  its  weight  =  35  —  1 5  =  20  g. 

Buoyancy  upon  the  body  =  40  +  20  =  60  g. 

Specific  gravity  of  the  body  =  40  -<-  60  =  .667 

Laboratory  Exercise  7. 

36.  Specific  Gravity  of  Liquids.  —  By  the  Specific  Gravity  Bottle. 
—  The  masses  of  equal  volumes  of  any  liquid  and  of  water  can  be 
obtained  by  finding  the  weight  of  each  that  will  completely  fill 
the  same  bottle.  The  bottle  is  weighed  when  empty,  when  filled 
with  the  liquid,  and  when  filled  with  water.  The  specific  gravity 
is  computed  from  these  three  weights.  A  bottle  having  a  glass 
stopper  should  be  used,  as  a  cork  or  a  rubber  stopper  would  prob- 
ably not  be  inserted  the  same  distance  each  time. 

By  Buoyancy  upon  a  Sinker.  — The  masses  of  equal  volumes  of 
any  liquid  and  of  water  can  be  obtained  by  finding  the  buoyant 
force  exerted  by  each  upon  the  same  solid  when  completely  im- 
mersed. The  sohd  is  weighed  in  air,  then  in  the  liquid,  then  in 
water.  The  solid  must  be  dense  enough  to  sink  in  either  liquid. 
A  glass  stopper  will  serve. 

By  Buoyancy  upon  a  Floating  Body:  the  Hydrometer.  —  The 
same  body  displaces  equal  weights  of  all  liquids  in  which  it  floats. 


Specific  Gravity 


29 


(Why  ?)  The  weight  of  the  displaced  Hquid  is  equal  to  the  prod- 
uct of  its  volume  and  density.  Let  v  and  V  denote  the  volumes 
of  two  liquids  displaced  by  the  same  floating 
body,  and  d  and  D  their  densities  respectively. 
Then  vd  =  VD  ;  whence  d'.DwVw.  That 
is,  the  volume  of  liquid  displaced  by  a  floating 
body  is  inversely  proportional  to  the  detisity  of 
the  liquid.  Thus,  if  the  second  liquid  is  twice 
as  dense  as  the  first,  the  floating  body  will  dis- 
place half  the  volume  of  it  that  it  does  of  the 
other. 

This  principle  is  utilized  in  the  hydrometer 
(Fig.  16).  It  consists  of  a  closed  glass  tube, 
with  a  bulb  at  the  lower  end  filled  with  shot  or 
mercury  to  keep  the  hydrometer  upright,  and  a 
second  bulb  farther  up,  the  purpose  of  which  is 
to  cause  the  greater  part  of  the  displacement. 
The  hydrometer  is  provided  with  a  paper  scale 
inclosed  within  the  tube,  and  graduated  so  that 
its  reading  at  the  surface  of  any  licjuid  in  which 
it  floats  is  the  specific  gravity  of  the  liquid.  The  reading  of  the 
scale  increases  toward  the  bottom.     (Why  ?) 

There  are  several  instruments  in  common  use  similar  to  the 
hydrometer,  but  having  scales  which  are  graduated  with  reference 
to  the  special  use  for  which  each  instrument  is  intended.  The 
alcoholi?neter,  for  determining  the  percentage  of  alcohol  in  liquors, 
and  the  lactometer^  for  determining  the  purity  of  milk,  are  exam- 
ples. 

Laboratory  Exercises  8  and  g. 


Fig.  16. 


PROBLEMS 

1.  A  body  weighs  500  g.  in  air  and  300  g.  in  water.     What  is  its  specific 
gravity  ? 

2.  A  stone  weighs  4CX)  g.  ;    its  density  is  2.5  g.  per  ccm.     What  will  it 
weigh  in  water  ? 


30  The  Mechanics  of  Liquids 

3.  A  body  weighs  200  g.  in  air  and  150  g.  in  water.     What  will  it  weigh 
in  a  salt  solution  the  specific  gravity  of  which  is  i.i  ? 

4.  Find  the  specific  gravity  of  a  solid  from  the  following  data  :  — 

Weight  of  solid  in  air  =15  g. 

Weight  of  sinker  in  water  =  35  g. 

W'eight  of  solid  and  sinker  in  water  =    5  g, 

5.  A  body  weighs  600  g.  in  air  and  250  g.  in  a  liquid  whose  specific  grav* 
ity  is  1.3.     Find  its  density. 

6.  A  stone  weighs  1000  lb. ;  its  volume  is  6  cu.  ft.     What  is  its  specific 
gravity  ? 

7.  A  body  weighs  600  g.  in  air,  300  g.  in  water,  and  250  g.  in  a  certain 
liquid.     What  is  the  sj>ecific  gravity  of  the  liquid  ? 

8.  A  body  weighs  300  g.  in  air  and   140  g.  in  alcohol,  the  specific  gravity 
of  which  is  .82.     What  is  the  volume  of  the  body  ? 

9.  Find  the  specific  gravity  of  a  litjuid  from  the  following  :  — 

Weight  of  bottle  =    40  g. 

Weight  of  Iwttle  filled  with  water  =  120  g. 

Weight  of  bottle  filled  with  the  licjuid  =  150  g. 

10.  A  piece  of  iron  weighs  1000  g. ;  its  specific  gravity  is  7.2.     What  will 
it  weigh  in  water  ? 

1 1.  A  stone  weighs  25  lb.  in  air  and  16  lb.  in  water.     What  is  its  spe- 
cific gravity  ? 

12.  Find  the  weight  of  a  cubic  foot  of  lead  from  its  specific  gravity. 


CHAPTER   III 

THE  MECHANICS  OP  GASES 

I.   Atmospheric  Pressure 

37.  Weight  of  Air.  —  A  vessel  full  of  air  is  appreciably  heavier 
than  the  same  vessel  after  the  air  has  been  partially  removed  by 
means  of  an  air  pump  {Exp.).  The  difference,  although  small, 
is  very  noticeable  if  the  vessel  is  of  considerable  size  (a  liter  or 
more)  and  the  weighing  is  done  on  a  sensitive  balance.  This 
proves  that  air  has  weight.  By  accurate  measurement  (the  neces- 
sary allowance  being  made  for  the  portion  of  the  air  remaining 
after  exhaustion),  the  weight  of  a  liter  of  air  at  ordinary  tempera- 
tures is  found  to  be  very  nearly  1.2  g.  (  =  .0012  g.  per  ccm.). 

38.  Pressure  of  the  Atmosphere. — The  weight  of  the  atmos- 
phere causes  it  to  exert  pressure,  since  each  horizontal  layer  sus- 
tains the  weight  of  all  the  air  above  it  and  adds  its  own  weight  to 
the  pressure  transmitted  to  the  next  lower  layer.  It  is  a  curious 
fact  that,  although  the  pressure  of  the  atmosphere  amounts  to  a 
total  of  about  35,000  pounds  on  a  person  of  average  size,  we  are 
unconscious  of  its  existence  under  the  ordinary  circumstances 
of  life.  The  existence  of  atmospheric  pressure  is  proved  by  the 
following  experiments. 

39.  Experiments  proving  the  Existence  of  Atmospheric  Pres- 
sure. —  A  sheet  of  thin  rubber  is  tied 

over  the  top  of  an  open  receiver  of 
an  air  pump  (Fig.  17).  As  the  air  is 
exhausted  from  the  receiver,  the  rub- 
ber is  depressed  more  and  more,  and 
soon  bursts  with  a  loud  report,  caused 
by  the  sudden   entrance  of  the   air 

3» 


32 


The  Mechanics  of  Gases 


{Exp.).  Before  the  air  is  exhausted,  it  exerts  a  pressure  equal 
to  that  of  the  outside  air,  and  the  rubber  remains  flat.  With 
each  stroke  of  the  pump  some  air  is  removed,  and  the  pressure  of 
the  remaining  air  is  correspondingly  diminished.  The  rubber  is 
therefore  pushed  inward  by  the  greater  pressure  of  the  atmosphere. 
If  the  hand  be  placed  over  the  receiver  instead  of  the  rubber, 
it  can  be  removed  only  with  difficulty  after  a  few  strokes  of  the 
pump,  so  firmly  will  it  be  held  in  place  by  the  pressure  of  the  air 
{Exp.).  The  flesh  of  the  pahii  will  be  distended  into  the  re- 
ceiver, and  will  seem  to  be  pulled  in.  This  sensation  is  entirely 
deceptive ;  the  flesh  is  pushed  in  by  the  pressure  of  the  liquids 
and  gases  within  the  body.  Since  the  pressure  within  the  body 
sustains  and  balances  the  pressure  of  the  atmosphere,  the  two 
must  be  equal.  Hence,  when  the  external  pressure  upon  any 
part  of  the  body  is  diminished,  that  part  will  be  distended  by  the 
greater  pressure  from  within. 

Two  hollow  brass  hemispheres  (Fig.  i8),  fitting  air-tight,  can 
easily  be  pulled  apart  so  long  as  none  of  the  air  has  been  removed 

from  the  space  they  inclose ;  but  after 
a  considerable  portion  of  this  air  has 

rated  only  with  great  difficulty  (/t;f/.). 

Before  the  air  is  exhausted  the  pres- 
FiG.  18.  sure   from  within    is    equal    to   that 

without  and  balances  it;  hence  the 
hemispheres  are  separated  without  hindrance  from  atmospheric 
pressure.  The  removal  of  a  portion  of  the  air  diminishes  the  pres- 
sure from  within  proportionally,  and  the  pressure  upon  the  outside 
pushes  the  hemispheres  together  with  great  force.  This  apparatus 
was  invented  about  the  middle  of  the  seventeenth  century  by  Otto 
von  Guericke,  burgomaster  of  Magdeburg,  Germany.  The  large 
copper  globe  that  he  used  for  the  first  trial  was  not  strong  enough 
to  withstand  the  great  pressure  upon  it,  and  suddenly  collapsed 
with  a  loud  report,  terrifying  all  the  spectators.  The  apparatus  is 
still  known  as  the  Magdeburg  hemispheres. 


'^1^ 


Atmospheric  Pressure  33 

40.  Laws  of  Gas  Pressure.  —  Pascal's  law  (Art.  28)  holds  for 
gases  as  well  as  for  liquids,  and  for  the  same  reason  —  their 
mobility. 

The  equal  pressure  of  the  atmosphere  in  all  direc- 
tions can  be  shown  with  the  apparatus  represented  in 
Fig.  19.  Thin  sheet  rubber  is  fastened  over  the  large 
end  of  a  thistle  tube,  and  a  rubber  tube  is  attached 
to  the  other  end.  On  exhausting  some  of  the  air 
with  the  mouth,  the  rubber  is  pushed  in  by  atmos- 
pheric pressure ;  and,  for  a  given  exhaustion  of  the 
tube,  the  depression  of  the  rubber  remains  the  same  in  whatever 
direction  the  tube  is  turned. 

The  law  of  proportionality  between  depth  and  pressure  (Law 
III,  Art.  23)  does  not  hold  for  the  atmosphere,  since  its  density 
increases  rapidly  with  the  depth,  gases  being  very  compressible. 
The  height  to  which  the  atmosphere  extends  is  unknown,  but  is 
variously  estimated  at  from  one  hundred  to  two  hundred  miles. 
It  is  known  to  extend  above  fifty  miles ;  yet  the  density  decreases 
so  rapidly  with  increase  of  altitude  that  the  pressure  at  a  height 
of  3.4  miles  is  only  one  half  as  great  as  at  sea  level. 
From  this  we  know  that  one  half  of  the  atmosphere  lies 
below  an  elevation  of  3.4  miles  above  sea  level.  Men 
have  ascended  to  higher  altitudes  than  this  upon  moun- 
tains and  in  balloons.  If  the  atmosphere  were  of  the  same 
density  throughout  as  at  sea-level,  it  would  extend  only 
to  a  height  of  about  five  miles. 

41.  Measurement  of  Atmospheric  Pressure.  —  Figure  20 
represents  a  tall  U-tube  about  half  full  of  mercury.  The 
mercury  stands  at  the  same  level  in  the  two  arms  under 
the  action  of  its  weight  and  the  equal  pressure  of  the  air 
upon  its  two  surfaces.  If  a  rubber  tube  be  attached  to 
one  arm  and  some  of  the  air  exhausted  by  means  of  an 
air  pump  or  by  applying  the  mouth,  the  pressure  in  that 
arm  will  be  diminished  and  the  greater  pressure  of  the 
atmosphere   in   the  open   arm   will  force   the  mercury  down  in 


34 


The  -Mechanics  of  Gases 


Fig.  ai. 


that   arm   and   up    in   the   other   until    equilibrium    is    restored 
^^^^  (Fig.  2i)  {£x/>.).     When  the  mercury  comes  to 

^rTfl  rest,  the  portion  of  it  below  the   level  ac  is  in 

m    *  equilibrium  under  the  action  of  the   downward 

m  pressure  of  the  atmosphere  on  the  surface  a  and 

■  the  equal  downward  pressure  at  c.     This  pressure 

I  at  r  is  the  transmitted  pressure  of  the  air  remain- 

■  ing  above  d,  increased  by  the  pressure  due  to  the 

weight  of  the  mercury  column  //c.  Hence  the 
pressure  due  to  the  weight  of  the  column  measures 
the  difference  between  the  pressure  of  the  atmos- 
phere and  the  pressure  of  the  air  remaining  in  the 
closed  arm.  For  example,  when  the  dilference 
of  level  of  the  two  surfaces  is  lo  cm.,  the  pressure 
of  the  atmosphere  exceeds  the  pressure  in  the 
closed  arm  by  lo  x  13.6  or  136  g.  per  scm. 
If  all  the  air  were  removed  from  the  closed  arm,  there  would 
be  no  pressure  upon  the  surface  of  the  mercury  in  it ;  and  the 
mercury  would  rise  in  this  arm  until  the  pressure  at  r, 
due  to  the  weight  of  the  column  /fc,  was  equal  to  the  J 
pressure  of  the  atmosphere.  This  important  fact  is  util- 
ized in  the  mercury  barometer,  an  instrument  for  measur- 
ing atmospheric  pressure.  Figure  22  represents  a  siphon 
barometer,  so  called  from  its  shape.  The  straight  part  of 
the  tube  is  80  cm.  or  more  in  length  and  its  upper  end  is 
sealed.  It  is  first  completely  filled  with  mercury  so  as  to 
expel  all  the  air.  When  turned  into  an  upright  position, 
the  mercury  falls  until  it  reaches  a  position  of  equilibrium, 
leaving  an  empty  space  {vacuum)  at  the  top.  The  differ- 
ence of  level  of  the  columns,  he,  is  called  the  hei^t^ht  of 
the  barometer.  This  height  multiplied  by  the  density  of 
mercury  measures  the  pressure  (per  scm.)  of  the  column 
at  c ;  and  consequently  measures  the  equal  pressure  of  the 
atmosphere.  Thus,  when  the  height  of  the  barometer  is  76  cm., 
the  atmospheric  pressure  is  76  x  13.596,  or  1033.3  g.  per  scm. 


y 


Fig.  22. 


Atmospheric  Pressure 


35 


^Z' 


\ 


Another  form  of  barometer  is  represented  in  Fig. 
straight  tube  about  85  cm.  in 
lenglh  and  closed  at  one  end,  is 
completely  filled  with  mercury, 
and,  with  a  finger  held  tightly 
over  the  open  end,  is  inverted,  and 
the  open  end  inserted  in  a  cup 
of  mercury.  The  height  of  the 
column  is  measured  from  the  sur- 
face of  the  mercury  in  the  cupt 
(Why  ?)  9  ?  J 

Atmospheric  pressure  is  gener- 
ally expressed  in  terms  of  the 
height  of  the  barometer,  measured 
in  centimeters  or  inches;  as  a 
pressure  of  75.3  cm.  {of  mercury) 
is  understood.  The  pressure  of 
the    atmosphere,    in    grams   per  l-i,: 

square  centimeter,  is  equal  to  the 

weight  of  a  vertical  column  of  air  one  square  centimeter  in  cross- 
section,  extending  from  the  place  where  the  pressure  is  taken  to 
the  upper  limit  of  the  atmosphere. 

Laboratory  Exercises  JO  and  li. 

42.  The  Barometer.  —  The  barometer  was  invented  by  Evange- 
lista  Torricelli  (1608-1647),  an  Italian  mathematician  and  scientist, 
in  1643,  some  years  before  Guericke's  celebrated  experiments  at 
Magdeburg.  The  space  above  the  mercury  in  a  barometer  is 
called  a  Torricellian  vacuum  in  honor  of  the  inventor. 

That  the  barometer  column  is  sustained  by  the  pressure  of  the 
atmosphere  was  conclusively  proved  by  Pascal,  under  whose  direc- 
tion the  height  of  the  barometer  was  determined  at  the  foot  and 
at  the  summit  of  Puy  de  Dome,  a  high  mountain  in  France.  The 
height  of  the  mercury  fell  nearly  8  cm.  during  the  ascent  of  about 
900  m.  The  effect  of  decrease  of  atmospheric  pressure  upon  the 
height  of  the  barometer  can  be    more    conveniently  shown   by 


23- 


36  The  Mechanics  of  Gases 

exhausting  the  air  from  a  tall  receiver  under  which  a  barometer 
tube  has  been  placed.  The  mercury  continues  to  fall  as  long  as 
the  process  of  exhaustion  is  continued  {£xp.). 

Barometer  tubes  are  mounted  in  a  variety  of  ways,  and  pro- 
vided with  scales  and  other  devices  for  convenience  and  accuracy 
in  reading.  Before  the  tube  is  mounted,  the  mercury  in  it  is 
boiled  to  expel  all  air  and  moisture. 

43.  Uses  of  the  Barometer.  —  It  is  a  well-established  fact  that 
different  conditions  of  the  weather  are  accompanied  or  preceded, 
with  considerable  regularity,  by  certain  changes  of  atmospheric 
pressure,  as  determined  by  the  height  of  the  barometer.  (Gener- 
ally speaking,  the  barometer  is  low  in  stormy  weather  and  high  in 
clear  weather ;  hence  the  approach  of  a  storm  is  indicated  by  a 
fall,  and  the  approach  of  Hiir  weather  by  a  rise,  of  the  barometer. 
The  difference  between  high  and  low  barometer  rarely  exceeds 
two  or  three  centimeters. 

This  knowledge  is  used  in  predicting  changes  in  the  weather ; 
but  the  forecasts  made  by  the  Weather  Bureau  are  based  on  other 
sources  of  information  as  well,  including  temperature,  direction 
and  velocity  of  the  wind,  the  course  and  progress  of  storms  up 
to  the  time  when  the  forecast  is  made,  and  the  existing  state 
of  the  weather ;  all  of  which  are  reported  to  the  central  office  by 
the  different  stations  distributed  over  the  country.  The  problem 
of  forecasting  the  weather  is  thus  a  very  complex  one. 

The  barometer  is  also  used  for  measuring  altitudes.  The 
change  of  pressure  due  to  a  given  change  of  altitude  being  known, 
the  height  of  a  mountain  can  be  computed  from  the  reading  of  a 
barometer  at  its  base  and  at  its  summit.  The  height  to  which  a 
balloon  ascends  is  determined  in  the  same  way.  For  moderate 
altitudes  above  sea  level,  it  is  approximately  correct  to  compute  the 
change  of  altitude  at  the  rate  of  900  ft.  for  a  fall  of  the  barometer 
of  one  inch. 

PROBLEMS 

1.  Explain  the  process  of  drinking  through  a  straw. 

2.  When  the  mercury  barometer  stands  at  a  height  of  76  cm.,  what  will 


Atmospheric   Pressure  37 

be  the  height  of  a  barometer  the  liquid  in  which  has  a  specific  gravity  of 
1.6? 

3.  When  the  barometer  stands  at  76  cm.,  a  hter  of  air  at  0°  C.  weighs 
1.293  g.  At  the  same  temperature  and  pressure,  what  will  be  the  weight  of 
the  air  in  a  room  9  m.  by  7  m.  and  4  m.  high  ? 

4.  Compute  the  weight  of  i  cu.  ft.  of  air  at  0°  C.  and  76  cm.  pressure  (sp. 
gr.  of  air  =  .001293). 

5.  What  weight  of  air  at  this  temperature  and  pressure  is  contained  in  a 
room  20  by  30  ft.,  and  12  ft.  high  ? 

6.  In  ascending  a  mountain  will  the  fall  of  the  barometer  during  each 
thousand  feet  of  ascent  be  greater  or  less  than  for  the  preceding  thousand 
feet?     Why  ? 

7.  (a)  The  weight  of  the  atmosphere  is  equal  to  the  weight  of  an  ocean 
of  mercury  covering  the  entire  surface  of  the  earth  to  what  depth  ?  (/v)  What 
would  be  the  depth  of  water  covering  the  entire  surface  of  the  earth  and 
having  equal  weight  ? 

8.  The  surface  of  the  body  of  a  man  of  medium  size  is  about  16  sq.  ft. 
Assuming  this  value  and  also  that  the  pressure  of  the  atmosphere  is  14.7  lbs. 
per  sq.  in.,  compute  the  total  pressure  that  a  man  sustains  upon  the  surface 
of  his  body. 

II.   Boyle's  Law 

44.  The  Elastic  Force  of  Gases.  —  Let  A  (Fig.  24)  represent 
any  portion  of  the  air.  It  may  be  thought  of  as  distinct  from  the 
surrounding  air,  although  no  bounding  sur- 
face actually  exists.  The  surrounding  air 
presses  inward  upon  all  sides  of  A,  as  indi- 
cated by  the  arrows  pointing  inward.  These 
pressures  are  balanced  at  every  point  by  an 
equal  outward  pressure  exerted  by  A  upon 
the  surrounding  air  (equal  action  and  reac-  P^^'  ^ 

tion).  This  pressure  exerted  by  A  is  not  the 
result  of  its  weight,  but  of  its  tendency  to  expand;  just  as  a  com- 
pressed spiral  spring  or  a  compressed  piece  of  rubber  exerts  an 
outward  pressure  that  is  independent  of  its  weight.  IVAy  gases 
tend  to  expand  is  a  question  that  will  be  considered  later  (Art.  182). 
The  fact  may  be  accepted  for  the  present  without  explanation. 

The  pressure  (per  unit  area)   exerted  by  any  body  of  gas  is 


38 


The  Mechanics  of  Gases 


called  its  elastic  force.  The  English  physicist,  Robert  Boyle,  who 
was  among  the  first  to  study  the  mechanics  of  the  air,  called  this 
elastic  force  the  spring  of  the  air.  The  appropriateness  of  the 
term  will  be  evident  if  one  suddenly  pushes  down  the  piston  of  a 
small  compression  pump  (such  as  a  bicycle  pump),  at  the  same 
time  keeping  the  tube  closed  to  prevent  the  escape  of  the  air 
{Exp.),  The  pressure  exerted  by  the  confined  air  rapidly  increases 
as  the  piston  is  pushed  farther  in,  and  this  pushes  the  piston  back 
again  when  it  is  released.  In  fact,  the  confined  air  acts  as  a 
spring  would  if  put  in  its  place.  The  experiment  shows  that  the 
elastic  force  of  a  gas  is  increased  by  compression. 

At  the  same  temperature  and  density,  the  elastic  force  of  the 
air  in  a  closed  vessel  is  equal  to  that  of  the  atmosphere.  (Which 
of  the  preceding  experiments  of  this  chapter  have  shown  this  to  be 
true?) 

45.  Measurement  of  the  Elastic  Force  of  Gases.  —  A  pressure 
gauge,  or  manometer ^  is  an  instrument  for  measuring  the  elastic 
force  of  a  gas  in  a  closed  space. 

An  Open  Manometer  (Fig.  25)  is  commonly  used  for  the  meas- 
urement of  pressures  only  slightly  greater  or  less 
^0j^  than  one  atmosphere.     It  consists  essentially  of  a 

^T  glass  U-tube  partly  filled  with  water  or  mercury, 

K  t,  with  a  rubber  tube  attached  to  one  arm  for  mak- 

■  ing  connections,  and  a  scale  for  measuring  the 

height  of  the  li(|uid  in  the  two  arms.  On  con- 
necting such  a  manometer  with  the  gas  pipes  and 
turning  on  the  gas,  the  liquid  will  be  pushed 
down  in  the  arm  in  which  the  gas  is  admitted. 
The  pressure  of  the  gas  upon  the  surface  a  is 
equal  to  the  pressure  at  the  same  level,  c,  in  the 
other  arm  ;  and  the  latter  pressure  is  the  sum  of 
the  atmospheric  pressure  upon  d,  and  the  pressure 
due  to  the  weight  of  the  column  of  liquid  dc. 
Hence  the  pressure  due  to  the  weight  of  l>c  meas- 
ures the  dijference  between  the  pressure  of  the  gas  and  the  pressure 


Fig.  25. 


Boyle's  Law  39 

of  the  atmosphere.  For  example,  if  the  liquid  in  the  manometer 
is  water  and  the  difference  of  level  8  cm.,  the  pres- 
sure of  the  gas  exceeds  that  of  the  air  by  8  g.  per  scm. 
The  Closed  Manofneter.  —  A  short  siphon  barom- 
eter with  a  rubber  tube  attached  to  the  open  arm 
(Fig.  26)  is  used  to  measure  the  pressure  in  partially 
exhausted  vessels.  Since  there  is  no  air  or  other  gas 
in  the  closed  arm,  the  mercury  completely  fills  it 
when  under  atmospheric  pressure.  While  the  air  or 
other  gas  is  being  pumped  from  a  vessel  to  which  a 
closed  manometer  is  attached,  the  mercury  continues 
to  fill  the  closed  arm  for  some  time,  if  the  original  ^ 

pressure  was  more  than  sufficient  to  sustain  the  full 
height  of  the  column.     It  is  only  after  the  mercury  begins  to  fall 
that  the  difference  of  level  of  the  columns  measures  the  pressure. 

46.  Units  for  the  Measurement  of  Pressure.  —  The  pressure  of 
gases  may  be  measured  in  any  of  the  following  units  :  — 

In  grams  per  square  centimeter  or  pounds  per  square  inch. 

In  centimeters  or  inches  of  mercury  or  of  water. 

In  atmospheres.  An  atmosphere  is  the  pressure  of  a  column 
of  mercury  76  cm.  high.  This  unit  is  approximately  the  average 
pressure  of  the  atmosphere  at  sea  level.  It  is  constant  and  is  not 
to  be  confounded  with  the  actual  atmospheric  pressure,  which 
varies  from  day  to  day  and  is  different  at  different  altitudes. 

The  pupil  should  be  able  to  formulate  rules  for  finding  the 
value  of  a  given  pressure  in  each  of  these  units  when  its  value  in 
terms  of  any  one  of  them  is  known. 

47.  Boyle's  Law.  — The  English  physicist,  Robert  Boyle  (1627- 
1691),  discovered  a  simple  relation  between  the  volume  of  a  gas 
and  the  pressure  upon  it.  This  relation,  known  as  Boyle's  law, 
has  been  found  to  be  approximately  true  for  all  gases.  It  is  as 
follows :  — 

The  temperature  remaining  the  same,  the  volume  of  a  given  body 
of  gas  varies  inversely  as  the  pressure  upon  it. 

For  example,  if  the  pressure  upon  any  body  of  gas  is  doubled, 


40  The  Mechanics  of  Gases 

the  volume  of  the  gas  will  be  decreased  one  half;  or,  if  the  pressure 
is  reduced  to  one  fifth  of  its  original  value,  the  volume  will  be- 
come five  limes  as  great  as  at  first. 

If  the  volume  of  a  mass  of  gas  is  Kj  when  the  pressure  upon  it 
is  Pi  (g.  per  scm.)  and  K  when  the  pressure  is  F^y  then,  ac- 
cording to  the  law,  /j : /^ : :  K^ :  l\.  From  this  proportion  we 
obuin  Pxy\  =  PiVt'i  that  is,  at  a  constant  temperature,  the  prod- 
uct of  the  volume  of  a  given  body  of  gas  and  the  pressure  upon  it 
is  constant. 

It  has  been  found  that  Boyle's  law  is  not  perfectly  exact  for 
any  gas ;  but  the  departure  from  the  law  is  so  slight  that  it  can  be 
detected  only  by  very  accurate  measurement,  unless  the  pressure 
is  so  great  that  the  gas  is  near  the  point  of  condensation.  The 
law  does  not  hold  if  the  change  of  pressure  is  accompanied  by  a 
change  of  temperature  ;  for  a  rise  of  temperature  will  itself  cause 
an  increase  of  pressure  or  of  volume  {Exp,). 

Laboratory  Exercise  J  J, 

48.  The  Relation  between  the  Density  and  the  Pressure  of  a 
Gas.  —  It  follows  from  lioyle's  law  that  — 

The  temperature  remaining  the  same^  the  density  of  a  gas  is  pro- 
portional to  the  pressure  upon  it. 

Thus  if  the  pressure  upon  a  quantity  of  gas  is  increased  three- 
fold, its  volume  will  be  one  third  as  great  as  at  first ;  and,  since 
the  entire  mass  occupies  one  third  its  former  volume,  its  density 
will  be  three  times  as  great  as  at  first.  The  increase  of  the  elastic 
force  of  a  gas  with  increase  of  density  is  well  illustrated  by  the 
effect  of  the  air  in  a  bicycle  tire.  After  the  tire  is  fully  inflated,  a 
further  supply  of  air  causes  a  proportionate  increase  in  its  density; 
and,  as  is  well  known,  this  makes  the  tire  "  harder." 

Laboratory  Exercise  14, 

PROBLEMS 

I.  A  cubical  vessel  20  cm.  in  each  dimension  is  full  of  air  at  a  pressure  of 
one  atmosphere.  What  is  the  total  pressure  exerted  by  the  confined  air  upon 
the  walls  of  the  vessel  ? 


Applications  of  Atmospheric  Pressure     41 

2.  Does  the  vessel  support  this  pressure  when  it  is  surrounded  by  air 
under  equal  pressure  ? 

3.  What  would  be  the  total  pressure  tending  to  burst  the  vessel  if  it  were 
placed  under  a  receiver  from  which  half  the  air  was  exhausted  (none  of  the 
air  being  removed  from  the  vessel)  ? 

4.  Find  the  weight  of  air  contained  in  the  vessel,  assuming  that  its  den- 
sity is  .(X)i2  g.  per  ccm. 

5.  {a)  At  what  depth  in  fresh  water  is  the  pressure  due  to  its  weight  equal 
to  one  atmosphere  ?     {b)  At  what  depth  in  salt  water  ? 

6.  From  what  depth  in  water  must  a  bubble  of  gas  start  in  order  that  its 
volume  may  be  doubled  on  reaching  the  surface  ? 

7.  {a)  What  is  the  pressure  in  pounds  per  square  inch  at  a  depth  of  3 
miles  in  the  ocean  ?  {b)  What  is  the  total  pressure  at  that  depth  upon  a  fish 
the  surface  of  whose  body  has  an  area  of  25  sq.  in.  ? 

8.  A  cubic  decimeter  of  gas  is  under  a  pressure  of  100  cm.  of  mercury. 
What  will  be  its  volume  at  the  same  temperature  under  a  pressure  of  30  cm. 
of  mercury  ? 

9.  A  liter  of  gas  is  taken  under  a  pressure  of  one  atmosphere.  What  will 
be  its  volume  at  the  same  temperature  under  a  pressure  of  100  cm.  of  mercury? 

10.  Two  liters  of  gas  under  a  pressure  of  one  atmosjihere  will  have  what 
volume  when  the  pressure  is  reduced  to  900  g.  per  scm.  ? 


III.  Applications  of  Atmospheric  Pressure 

49.  The  Air  Pump.  —  A  simple  form  of  air  pump  is  represented 
in  Fig.  27.  The  pump  consists  of  a  metal  cylinder  in  which  fits 
an  air-tight  piston  operated  by  the  handle.  There  are  two  valves, 
a  and  bj  the  former  in  the  piston  and  the  latter  at  the  entrance  of 
the  tube,  at  the  bottom  of  the  cylinder.  The  valves  open  in  one 
direction  only,  as  shown  in  the  figure.  The  simplest  form  of  valve 
consists  of  a  piece  of  flexible  leather,  placed  so  as  to  cover  the 
hole  and  fastened  at  one  edge.  The  valve  closes  the  opening  air- 
tight when  pressed  against  it,  and  leaves  it  open  when  pushed  in 
the  opposite  direction.  The  pump  is  connected  by  a  tube  to  an 
opening,  (9,  at  the  center  of  a  flat  metal  plate,  PQ,  upon  which 
stands  a  receiver,  R. 

Suppose  the  piston  to  be  at  rest  at  the  bottom  of  the  cylinder. 
Both  valves  will  be  closed,  being   held   down   by    their   weight. 


42 


The  Mechanics  of  Gases 


During  the  up  stroke  of  the   piston,  the   small   amount  of  air 
beneath  it  expands  and  fills  the  increased  space,  and  its  pressure 


Fig.  27. 

decreases  proportionally.  The  atmospheric  pressure  upon  the 
top  of  valve  a  being  now  greater  than  the  pressure  from  beneath, 
this  valve  is  firmly  closed.  When  the  downward  pressure  upon  d 
is  sufficiently  diminished,  the  pressure  of  the  air  in  the  tube 
beneath  this  valve  lifts  it,  permitting  some  of  the  air  in  the 
receiver  to  escape  into  the  space  below  the  piston.  As  soon  as 
the  piston  stops  rising,  the  lower  valve  is  closed  by  its  own  weight. 
On  pushing  the  piston  down,  the  air  beneath  it  is  compressed. 
This  air  cannot  escape  through  the  lower  valve,  since  the  increased 
pressure  only  closes  this  valve  more  tightly.  When  the  amount  of 
compression  is  such  that  the  density  of  the  confined  air  is  slightly 
greater  than  that  of  the  atmosphere,  the  upper  valve  is  forced 
open,  permitting  the  air  to  escape. 

These  processes  are  repeated  with  every  stroke  of  the  piston, 
thus  gradually  removing  the  air  from  the  receiver.    The  limit  of 


Applications  of  Atmospheric   Pressure     43 


possible  exhaustion  is  reached  when  the  pressure  of  the  air  re- 
maining in  the  receiver  is  insufficient  to  Hft  the  lower  valve,  or 
when  the  quantity  of  air  that  enters  the  cylinder  with  the  up  stroke 
is  so  small  that  it  cannot  be  compressed  enough  to  raise  the  upper 
valve. 

Pumps  for  obtaining  a  more  nearly  perfect  vacuum  are  pro- 
vided with  metal  valves  or  stoj)cocks,  operated  automatically  by 
a  simple  mechanism  attached  to  the  piston  or  to  the  piston 
rod. 

50.  The  Compression  Pump.  —  If  the  valves  of  the  pump  repre- 
sented in  Fig.  27  were  made  to 
open  in  the  opposite  direction, 
the  pump,  when  operated,  would 
force  air  into  the  receiver.  A 
pump  made  to  force  air  or  any 
gas  into  a  closed  vessel  is  called 
a  compression  pump.  A  pump 
such  as  is  represented  in  Fig.  28 
may  be  used  either  for  exhaust- 
ing or  compressing  gases.  On 
operating  the  pump,  air  enters  it 
through  A  and  leaves  it  through 
C.     Hence  if  a  closed  vessel  be 

attached  to  C,  air  will  be  forced  into  it ;  if  attached  to  Ay  the  air 
will  be  exhausted  from  it. 

A  bicycle  pump  is  a  compression  pump  of  very  simple  construc- 
tion. It  has  but  one  valve,  the  entire  piston  serving  this  purpose. 
The  valve  in  the  tube  of  the  bicycle  tire 
takes  the  place  of  the  outlet  valve  in  the 
pump.  (Examine  a  bicycle  pump  and 
explain  its  action.) 

A  bellows  is  a  form  of  compression 
pump.  It  is  provided  with  two  valves, 
a  and  b  (Fig.  29),  the  former  opening  inward,  the  latter  outward. 
(Explain  its  action.) 


Fig.  28. 


Fig.  29. 


44 


The  Mechanics  of  Gases 


51.  The  Lifting  Pump.  —  The  lifting  or  suction  pumpy  used  for 
pumping  water,  is  similar  to  an  air  pump  in  its  construction  and 
action.  The  valves  open  upward,  as  shown  in  the  figure.  A  pipe 
extends  from  the  cylinder  or  barrel  of  the  pump  to  some  distance 
below  the  surface  of  the  water  in  the  well  or  cistern.  The  piston 
is  operated  by  means  of  a  lever,  called  the  handle.  Starting  with 
the  pump  "  empty,"  it  first  acts  as  an  air  pump  to  exhaust  the  air 
from  the  pipe  (see  Art.  49).  During  this  process  the  pressure  of 
the  air  within  the  barrel  and  the  pipe  decreases  and  the  greater 
pressure  of  the  air  upon  the  water  in  the  well  forces  some  of  it  up 


Fig.  30. 


Fig.  31. 


Fig.  32. 


into  the  pipe ;  the  pressure  due  to  the  weight  of  the  column  of 
water  thus  sustained  being  equal  to  the  difference  between  the 
atmospheric  pressure  and  the  pressure  of  the  air  remaining  in  the 
pump. 

After  the  pump  is  filled  with  water,  the  water  below  the  piston 
follows  it  during  the  up  stroke,  being  pushed  upward  through  the 
lower  valve.  When  the  piston  begins  to  descend,  the  lower  valve 
closes,  preventing  the  return  of  the  water  into  the  pipe.  The 
valve  in  the  piston  is  forced  open  at  the  same  time,  and  the  water 
flows  through  it  into  the  space  above.     At  the  beginning  of  the 


Applications  of  Atmospheric  Pressure     45 

up  stroke,  the  valve  in  the  piston  falls  and  the  water  above  it  is 
lifted  out. 

Since  the  entire  pressure  of  the  atmosphere  can  sustain  a  col- 
umn of  water  only  to  a  height  of  about  10.3  m.  (34  ft.),  the  lower 
valve  would  have  to  be  within  that  distance  of  the  water  in  the 
well  even  if  the  pump  were  capable  of  producing  a  perfect  vacuum. 
The  actual  limit  of  distance  is  about  28  or  30  ft. 

52.  The  Force  Pump.  —  In  the  force  pump  the  second  valve  is 
placed  at  the  entrance  to  the  discharge  pipe,  B  (Fig.  32).  There 
is  no  valve  in  the  piston.  The  action  of  the  pump  during  the  up 
stroke  of  the  piston  is  the  same  as  in  the  lifting  pump.  (Which 
valve  is  open  ?  Which  closed  ?)  With  the  down  stroke  of  the 
piston  the  water  is  forced  into  the  discharge  pipe.  The  height  to 
which  water  can  be  forced  in  the  discharge  pipe  depends  only 
upon  the  strength  of  the  pump,  being  in  no  way  affected  by 
atmospheric  pressure. 

Force  pumps  are  generally  provided  with  an  air  chamber,  D, 
connected  with  the  discharge  pipe.  During  the  down  stroke  of 
the  piston  the  water  is  forced  into  the  chamber,  compressing  the 
air  above  it.  The  elastic  force  of  the  compressed  air  maintains 
the  flow  from  the  air  chamber  during  the  up  stroke  of  the  piston, 
making  the  flow  continuous.  The  force  pump  is  used  to  force 
water  to  considerable  heights,  and  to  deliver  it  under  great  pres- 
sure, as  in  fire  engines. 

53.  The  Siphon.  —  A  bent  tube  or  pipe  for  transferring  liquids 
over  an  elevation  from  a  higher  to  a  lower  level  is  called  a  siphon 
(Fig.  33).  Either  a  rigid  or  a  flexible  tube 
will  serve  the  purpose.  To  start  a  small 
siphon,  it  may  be  held  with  the  bend  down 
and  filled,  then,  with  a  finger  over  each 
end,  inverted  and  placed  in  position  ;  or  it 
may  be  placed  in  position  and  the  air  then 
exhausted  by  applying  the  mouth  to  the 
lower  end.  Siphons  are  generally  provided  with  a  suction  tube 
for  this  purpose,  so  that  the  liquid  will  not  flow  into  the  mouth. 


46  The  Mechanics  of  Gases 

The  liquid  will  continue  to  flow  as  long  as  one  end  of  the  siphon 
is  covered  by  it,  and  the  other  end  is  below  the  level  of  its  surface, 
/>.  below  ab  in  the  figure ;  but  if  the  outlet  of  the  siphon  is  also 
immersed,  the  flow  will  cease  as  soon  as  the  liquids  in  the  two 
vessels  reach  the  same  level. 

To  explain  the  action  of  the  siphon  we  may  suppose  it  to  be 
stopped  by  closing  the  outlet,  r,  with  the  finger.  The  liquid  will 
then  be  at  rest,  and  the  laws  of  pressure  for  liquids  in  equilibrium 
will  hold.  At  points  a  and  b  in  the  tube,  on  a  level  with  the  sur- 
face of  the  liquid,  the  pressure  is  the  same  as  that  of  the  atmos- 
phere. The  pressure  at  c  is  equal  to  this  plus  the  pressure  due 
to  the  weight  of  the  liquid  column  be.  Hence,  when  the  finger 
is  removed,  this  pressure  of  the  column  be  acts  as  an  unbalanced 
force  u|X)n  the  liquid  in  the  siphon,  causing  it  to  flow.  The  liquid 
is  held  in  a  continuous  column  by  the  pressure  of  the  atmosphere, 
acting  at  the  ends  of  the  siphon  ;  otherwise  the  liquid  would  part 
at  the  top  and  run  out  at  both  ends,  leaving  the  siphon  empty. 
It  is,  in  fact,  the  transmitted  pressure  of  the  atmosphere  that  forces 
the  liquid  up  the  short  arm. 

Laboratory  Rxereise  12. 

64.  Respiration.  —  In  breathing,  the  size  of  the  chest  cavity  is 
alternately  increased  and  diminished  by  muscular  action.  The 
pressure  of  the  air  in  the  lungs  causes  them  to  expand  so  as 
always  to  fill  the  space  afforded  them ;  hence,  when  the  chest  is 
raised  and  the  diaphragm  depressed  in  inhaling,  the  expansion  of 
the  air  already  in  the  lungs  diminishes  its  pressure  and  more  air  is 
pushed  into  the  lungs  by  the  greater  pressure  of  the  outside  air. 
The  familiar  expression  "  drawing  in  a  breath "  is  misleading  in 
that  it  implies  a  pulling  force.  When  the  chest  is  contracted  in 
exhaling,  the  air  in  the  lungs  is  compressed  and  some  of  it  is 
forced  out. 

55.  Buoyancy  of  the  Air.  —  A  body  of  considerable  size  and  of 
small  specific  gravity  weighs  appreciably  more  under  a  partially 
exhausted  receiver  than  it  does  in  air.  This  fact  may  be  illustrated 
with  the  apparatus  shown  in  Fig.  34.     A  hollow  globe,  closed  air- 


Applications  of  Atmospheric  Pressure     47 


FIG.  34. 


tight,  is  exactly  balanced  in  air  by  a  solid  brass  weight.     When 
the  apparatus  is  placed  under  the  receiver  of  an  air  pump  and  the 
air  exhausted,  the  globe  descends, 
showing  that  it  is  now  heavier  than 
the  weight  {Exp.). 

The  experiment  proves  that  air 
exerts  a  buoyant  force.  The  globe 
and  the  solid  body  have  equal 
weight  in  air  ;  but  the  buoyant  force 
of  the  air  is  greater  upon  the  globe, 
since  it  is  much  the  larger  of  the 
two.  Hence,  with  partial  exhaus- 
tion of  the  air  in  the  receiver,  there 
is  greater  loss  of  buoyancy  upon  the 
globe,  and  it  therefore  sinks.  The 
law  of  buoyancy  for  gases  is  the 
same  as  for  liquids  and  for  the  same  reasons  (Art.  31). 

The  amount  of  the  buoyant  force  of  the  air  upon  solids  and 
liquids  is  relatively  very  small,  and  in  the  affairs  of  daily  life  may 
be  disregarded.^  The  true  weight  of  a  body  is  its  weight  in  a 
vacuum  ;  its  weight  in  air  in  called  its  apparent  weight  when  it  is 
necessary  to  distinguish  between  the  two.  The  difference  between 
the  true  and  the  apparent  weight  of  a  body  is,  of  course,  the 
buoyant  force  of  the  air  upon  it.  The  buoyant  force  of  the  air 
upon  gases  is  relatively  large.  In  fact,  upon  gases  less  dense  than 
air  it  exceeds  their  true  weight.  Such  a  gas  tends  to  rise,  just  as 
a  cork  does  in  water.  The  weight  of  a  gas  is  always  understood 
to  mean  its  true  weight. 

56.  The  Balloon.  — A  balloon  is  sustained  by  the  buoyant  force 
of  the  air,  the  gas  with  which  it  is  filled  being  lighter  than  air. 
Hydrogen  is  best  adapted  to  the  purpose,  being  the  lightest  of 
gases  ;  but  illuminating  gas  is  generally  used,  as  it  is  cheaper  and 
more  easily  obtained.     Hot  air  was  used  in  the  balloons  first  in- 

1  The  buoyant  force  of  the  air  upon  i  ksj.  (i  liter)  of  water  is  the  weight  of  a 
liter  of  air,  or  about  1.2  g, ;  upon  i  kg.  of  lead  the  buoyant  force  is  about  .1  g. 


48 


The  Mechanics  of  Gases 


vented.  A  balloon  will  rise  if  the  buoyant  force  upon  it  is  greater 
than  its  true  weight,  including  the  weight  of 
the  gas  with  which  it  is  filled  and  the  weight  of 
the  car  and  its  load. 

A  balloon  is  not  fully  inflated  at  the  start, 
space  being  left  for  the  expansion  of  the  gas 
as  the  atmospheric  pressure  upon  it  dimin- 
ishes during  the  ascent.  As  long  as  this  ex- 
pansion continues,  the  buoyant  force  upon  a 
balloon  remains  constant,  for  the  increase  in 
the  volume  of  the  displaced  air  offsets  the  de- 
crease in  its  density.  As  a  balloon  rises  after 
becoming  fully  distended,  the  buoyant  force 
decreases  until  it  is  no  greater  than  the  true 
weight  of  the  balloon  and  all  it  carries.  The 
balloon  then  ceases  to  rise,  unless  lightened 
by  throwing  out  sand,  a  supply  of  which  is 
carried  for  that  purpose.     When  the  aeronaut 

wishes  to  descend,  he  opens  a  valve  at  the  top  of  the  balloon  and 

some  of  the  gas  escapes. 


Fig.  35. 


PROBLEMS 


1.  Orer  how  great  an  elevation  can  water  be  siphoned  ?  Why  ?  Over 
how  great  an  elevation  can  mercury  be  siphoned  ?  Would  a  siphon  work  in 
a  vacuum  ?     Explain. 

2.  (a)  At  ordinary  temperatures  and  under  a  pressure  of  one  atmosphere, 
a  cubic  meter  of  air  weighs  about  1.2  kg.,  a  cubic  meter  of  hydrogen  about 
.083  kg.,  and  a  cubic  meter  of  illuminating  gas  about  .74  kg.  Assuming 
these  values,  what  is  the  buoyant  force  upon  a  balloon  containing  500  cu.  m. 
of  hydrogen  ?  (d)  How  great  a  weight  will  this  buoyant  force  sustain 
in  addition  to  the  weight  of  the  hydrogen  ? 

3.  With  what  volume  of  illuminating  gas  must  a  balloon  be  filled  to  rise, 
if  the  empty  balloon,  the  car,  and  the  occupants  together  weigh  500  kg.  ? 

4.  Will  the  true  weight  of  a  body  be  greater  or  less  than  its  weight  in 
air  when  weighed  on  an  equal-arm  balance  with  brass  weights  (a)  if  the 
density  of  the  body  is  the  same  as  that  of  brass?  (^)  if  its  density  is  less? 
(<•)  if  its  density  is  greater  ? 


CHAPTER   IV 

STATICS   OP   SOLIDS 

I.   Concurrent  Forces 

57.  Mechanics.  —  Mechanics  is  the  branch  of  physics  that  treats 
of  the  action  of  forces  upon  bodies.  It  is  divided  into  statics  and 
dynamics  or  kinetics. 

Statics  is  the  mechanics  of  balanced  forces  (Art.  ii)  ;  it  treats 
of  the  relations  that  must  exist  among  the  forces  acting  upon  a 
body  at  rest  in  order  that  the  body  may  remain  at  rest.  The 
statics  of  fluids  is  the  subject  of  the  two  preceding  chapters.  The 
subject  of  the  present  chapter  is  the  statics  of  solids. 

Dynamics^  or  kinetics,  is  the  mechanics  of  unbalanced  forces 
(Art.  1 1 )  ;  it  treats  of  the  effects  of  unbalanced  forces  in  produc- 
ing and  changing  motion.  The  dynamics  of  solids  is  the  subject 
of  the  following  chapter. 

58.  Equilibrium  of  Two  Forces.  — The  relations  that  must  exist 
among  two  or  more  forces  in  order  that  they  may  balance  each 
other  are  called  the  conditions  necessary  for  equilibriutn^  or,  simply, 
the  conditions  of  equilibrium. 

The  conditions  of  equilibrium  for  two  forces  can  be  studied  by 
means   of  two   drawscales  and   a  board   supported   upon   three 


Fig.  36. 

marbles  lying  on  a  table  (Fig.  36).     Cords  are  attached  to  nails  at 
A  and  B.     Horizontal  forces  are  applied  to  the  board  through 

49 


50  Statics  of  Solids 

these  cords  and  are  measured  by  the  drawscales.  If  these  forces 
are  in  equiUbrium  with  each  other,  the  board  will  remain  at  rest  ; 
if  they  are  not  in  equilibrium,  it  will  move,  since  the  friction  is 
inappreciable.  By  trial  with  the  apparatus  it  will  be  found  that  : 
( I )  When  equal  forces  are  applied  in  opposite  directions  but  not 
along  the  same  line,  the  board  will  not  be  in  equilibrium,  but  will 
rotate  until  the  lines  of  action  of  the  forces  coincide  (Fig.  37). 


Fig.  37. 

The  board  will  then  be  in  equilibrium.  (2)  When  the  applied 
forces  are  opposite  and  have  the  same  line  of  action,  but  are 
unequal,  the  board  will  be  pulled  in  the  direction  of  the  greater 
force.  (3)  When  the  forces  are  either  equal  or  unequal  but 
not  opposite  in  direction,  the  board  will  not  be  in   equilibrium 

The  experiment  shows  that  fwo  forces  balance  each  other  only 
when  they  are  equal  in  magnitude^  opposite  in  direction^  and  have 
the  same  line  of  action. 

A  and  B  are  the  points  of  application  of  the  forces  respectively. 
A  force  produces  the  same  effect  when  it  is  applied  at  any  other 
point  in  the  same  line  of  action.  Thus,  if  either  of  the  equal  and 
opposite  forces  be  applied  at  C  (Fig.  37)  instead  of  at  A  or  B, 
they  will  still  be  in  equilibrium. 

59.  Elements  of  a  Force.  —  The  effect  of  a  force  depends  upon 
its  magnitude^  its  direction,  and  \\.%  point  of  application  (or  line  of 
action) .  These  are  called  the  elements  of  a  force.  They  must  all 
be  considered  in  describing  and  comparing  forces,  and  in  discuss- 
ing their  effects. 

60.  Representation  of  Forces.  —  In  studying  the  relations  of  a 
set  of  forces  to  one  another,  it  is  often  convenient  to  make  use  of 


>B 


Concurrent  Forces  51 

a  diagram  in  which  each  force  is  represented  by  a  line.  The 
direction  of  the  force  is  represented  by 
the  direction  of  the  line,  with  an  arrow- 
head placed  on  it  ;  the  magnitude  of  the 
force,  by  the  length  of  the  line;  and  its 
point  of  application,  by  the  point  from 
which  the  line  is  drawn.     The  method  is 

illustrated  in  Fig.  38,  which  represents  two  forces  having  a  common 
point  of  application,  O,  and  differing  in  direction  by  a  right  angle. 
The  force  represented  by  OB  is  twice  as  great  as  the  other. 

The  magnitude  of  a  force  can  be  represented  on  any  scale 
desired.  Thus  i  cm.  may  represent  a  force  of  10  g.,  100  g., 
500  g.,  etc.  But  the  same  scale  must  be  used  for  all  forces  in  the 
same  figure. 

61.  Resultant  and  Components.  —  In  many  cases  where  two  or 
more  forces  act  upon  a  body  at  the  same  time,  a  single  force  can 
be  found  which,  acting  alone,  would  produce  the  same  effect  upon 
the  body  as  the  given  forces.  This  one  force  is  called  the  result- 
ant of  the  forces  to  which  it  is  equivalent,  and  the  latter  are  called 
the  cojnponetits  {i.e.  parts)  of  the  resultant. 

The  resultant  of  any  number  of  forces  acting  along  the  same 
line  in  the  same  direction  is  their  sum.  Thus,  if  a  boy  pulls  on  a 
cart  with  a  force  of  15  lb.,  and  another  boy  pulls  with  him,  exerting 
a  force  of  25  lb.,  the  effect  upon  the  cart  will  be  the  same  as  that 
of  a  single  force  of  40  lb.  acting  in  the  same  direction. 

The  resultant  of  two  forces  acting  in  opposite  directions  along 
the  same  line  is  their  difference,  and  its  direction  is  that  of  the 
greater  component.  The  resultant  of  two  equal  forces  acting 
in  opposite  directions  along  the  same  line  is  zero,  since  the  two 
forces  exactly  neutralize  each  other  (Art.  58).  The  resultant  of 
any  set  of  balanced  forces  is  zero,  for  the  same  reason. 

The  process  of  finding  the  resultant  of  two  or  more  given  forces 
is  called  the  composition  of  forces.  In  the  case  of  forces  acting 
along  the  same  line,  composition  is  effected  by  adding  all  the  forces 
that  act  in  one  direction,  and  subtracting  all  that  act  in  the  oppo- 


52 


Statics  of  Solids 


site  direction.  Other  methods  are  required  for  forces  acting  at  an 
angle  or  along  different  parallel  lines,  as  shown  in  the  following 
articles. 

62.    Equilibrium  of  Three  Concurrent  Forces.  — ^  Forces  whose 

lines  of  action  meet  in  a  point  are 
called  concurrent  forces. 

A  simple  form  of  apparatus  for 
studying  the  conditions  of  equilibrium 
for  three  concurrent  forces  is  shown 
in  Fig.  39»  Three  cords  are  tied  to 
a  ring  and  a  drawscale  is  attached 
to  each.  The  scales  are  adjusted  so 
that  all  exert  a  considerable  force  at 
the  same  time.  The  ring  will  be  in 
equilibrium  under  the  action  of  the 
three  forces,  all  of  which  lie  in  the 


Fig.  39. 


same  plane.     These  forces  are  concurrent  at 
the  center  of  the  ring;   their  directions  are 
outward  from  this  center  in  the  directions  of 
the  cords ;  and  their  magnitudes  are  given  by 
the  readings  of  the  scales.     In  order  to  deter- 
mine the  relations  that  exist  among  the  forces, 
they  are  represented  in  magnitude  and  direc- 
tion  by  the  lines   <j,  b^  and   r,  respectively 
(Fig.  40).     When  the  experimental 
work  and  the  construction  are  ac- 
curate, it   will   be  found   that   the 
diagonal  R  of  the  parallelogram  con- 
structed upon  any  two  of  these  lines 
as  sides,  is  equal  to  the  third  line 
and  is  in  exactly  the  opposite  direction  from  O,    The  conditions 
necessary  for  equilibrium  may  therefore  be  stated  as  follows  :  — 

In  order  that  three  concurrent  forces  may  be  in  equilibrium, 
their  lines  of  action  must  lie  in  the  satne  plane,  and  the  magnitudes . 
and  directions  of  the  forces  must  be  such  that,  if  any  two  of 


Fig.  40. 


Concurrent  Forces  53 

the  lines  representing  them  be  taken  as  the  sides  of  a  parallelogram^ 
the  concurrent  diagonal  of  this  parallelogram  will  be  equal  and 
opposite  to,  the  line  representing  the  third  force. 
Laboratory  Exercise  75. 

63.  Resultant  of  Two  Concurrent  Forces :  Parallelogram  of 
Forces.  —  If  three  concurrent  forces  are  in  equilibrium,  the  result- 
ant of  any  two  of  them  must  be  equal  and  opposite  to  the  third ; 
for  they  together  balance  the  third  force,  and,  by  definition,  their 
resultant  is  the  single  force  that  would  produce  the  same  effect. 
Thus,  the  resultant  of  the  forces  represented  by  a  and  b  in  Fig. 
40  is  represented  by  the  diagonal  R^  for  we  have  seen  that  this 
diagonal  is  equal  and  opposite  to  c. 

This  relation  between  two  concurrent  forces  and  their  resultant 
is  known  as  the  parallelogram  of  forces.  It  may  be  stated  as 
follows  :  If  two  concurrent  forces  are  represented  by  lines  drawn 
from  the  same  pointy  the  concurrent  diagonal  of  the  parallelogram 
constructed  upon  these  lines  as  sides^  will  represent  their  resultant 
in  magnitude  and  direction.  The  numerical  value  of  the  resultant 
is  found  from  the  length  of  this  diagonal  by  applying  to  it  the 
scale  of  lengths  chosen  for  the  construction. 

The  resultant  of  two  concurrent  forces  can  always  be  found  by 
the  above  construction.  It  can  also  be  computed  by  the  rules  of 
trigonometry.  In  a  few  cases  the  resultant  can  be  computed  from 
the  relations  established  in  plane  geometry.  The  most  important 
case  is  that  of  two  forces  acting  at  an  angle  of  90°.  In  this  case 
the  resultant  is  equal  to  the  square  root  of  the  sum  of  the  squares 
of  the  components  (since  the  square  of  the  hypothenuse  of  £^  right 
triangle  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides). 

64.  Composition  of  More  than  Two  Concurrent  Forces.  —  The 
resultant  of  any  number  of  concurrent  forces  can  be  found  by 
combining  the  resultant  of  any  two  of  them  with  a  third,  their 
resultant  with  the  fourth,  and  so  on  till  each  force  has  been  in- 
cluded once  in  the  construction  or  computation,  The  last  result- 
ant is  the  resultant  of  all  the  forces. 


54 


Statics  of  Solids 


65.  Equilibrant. — The  single  force  that  would  balance  one  or 
more  given  forces  is  called  their  equilibrant.  The  equilibrant  of 
any  number  of  forces  is  equal  and  opposite  to  their  resultant. 
Either  of  two  forces  in  equilibrium  is  the  equilibrant  of  the  other ; 
and  any  one  of  three  forces  in  equilibrium  is  the  equilibrant  of  the 
other  two  ( Fig.  40) . 

66.  Resolution  of  a  Force.  —  It  is  frequently  necessary  in  study- 
ing the  effects  of  a  force  to  consider  it  as  being  replaced  by  two 
or  more  concurrent  forces  which  are  together  equivalent  to  it. 
The  given  force,  when  so  treated,  is  said  to  be  resolved  into  its 
components,  and  the  process  is  called  resolution.  It  is  the  reverse 
of  composition.  A  force  can  be  resolved  into  any  number  of  sets 
of  components  ;  but  resolution  into  perpendicular  components  is 
most  frequently  required,  as  in  the  following  problem. 

A  ball  whose  weight  is  IK  rests  upon  a  plane,  AB  (Fig.  41), 

inclined  at  an  angle  ^^Cwith  the 
horizontal.  What  portion  of  the 
weight  of  the  ball  tends  to  cause 
it  to  roll  down  the  plane  ?  The 
weight  of  the  ball  is  represented  by 
OW.  Its  actual  direction  is  verti- 
cally down ;  but  it  may  be  regarded 
as  being  made  up  of  two  compo- 
nents, OP  and  OF.  The  first,  acting  perpendicularly  to  the  plane, 
causes  pressure  upon  it ;  the  second,  acting  parallel  to  the  plane, 
will,  if  unbalanced,  cause  the  ball  to  roll  down  it.  A  force  equal 
and  opposite  to  OF  will  hold  the  ball  in  equilibrium. 

PROBLEMS 

NoTK.  —  The  numerical  results  called  for  in  the  following  problems  are  all  to  be  found 
by  computation. 

1.  A  weight  of  100  kg.  is  supported  by  two  cords  making  equal  angles 
with  the  horizontal  and  an  angle  of  120°  with  each  other. 
What  is  the  tension  on  each  cord  ? 

2.  What  would  be  the  tension  on  each  cord  supporting 
the  above  weight  if  one  made  an  angle  of  30°  with  the 
horizontal,  and  the  other  60°  ?  Fig.  4a. 


Fig.  41. 


Parallel  Forces 


55 


3.  What  would  be  the  tension  on  each  cord  if  each  made  an  angle  of  45° 
with  the  horizontal  ? 

4.  A  ball  is  placed  on  a  plane  inclined  at  an  angle  of  30°.  What  fraction 
of  its  weight  tends  to  cause  motion  down  the  plane  ? 

5.  If  the  weight  of  the  ball  in  the  previous  problem 
is  2  kg.,  what  pressure  does  it  exert  upon  the  plane  ? 

6.  A  body  weighing  30  kg.  is  suspended  by  a  cord  as 
shown  in  Fig.  43.    The  ends  of  the  cord  are  fastened 

at  points  A  and  B  at  the  same  height.  The 
portion  AC  ol  the  cord  is  40  cm.  long,  and 
the  portion  ^C  is  70  cm.  Angle  ACB  is  a 
right  angle.     What  is  the  tension  sustained  by  AC?  hy  BC ? 

7.  A  weight  of  120  g.  is  tied  to  a  cord  and  suspended  at  O 
(Fig.  44).  A  second  cord  is  attached  to  the  supporting  cord  at 
Ay  a  distance  of  100  cm.  from  6>,  and  is  pulled  in  a  horizontal 
direction  till  A  is  drawn  20  cm.  from  the  vertical  through  O, 
Find  the  tension  upon  AO  and  upon  AF. 

8.  Find   the  resultant  of  three   concurrent  forces  of  5,   16, 
and  II  lb.,  respectively,  the  first  two  acting  in  opposite  direc- 
tions, and  the  third  at  right  angles  to  them  (Fig.  45). 

9.  A  weight  of  ico  lb.  is  suspended  at  the 
middle  of  a  rope,  ACB  (Fig.  46),  20  ft.  long. 
The  ends  of  the  rope  are  fastened  at  points 
A  and  B  at  the  same  height.  What  is  the 
tension  of  the  rope  when  CD  is  3  ft.  ?  when  CD  is  I  ft.? 
when  CD  is  i  inch  ? 

10.  (a)  A  brick  lies  on  the  ground.  What  is  the  equilibrant  of  the  weight 
of  the  brick  ?  {b)  What  is  the  reaction  of  the  pressure  of  the  brick  upon 
the  ground  ? 

11.  How  does  the  reaction  of  a  force  differ  from  its  equilibrant  ?  Men- 
tion examples  to  illustrate. 

12.  If  any  number  of  forces  and  their  equilibrant  together  act  upon  a 
body,  what  is  their  combined  effect  ? 


r 

Fig.  45. 


II.   Parallel  Forces 

67.  Equilibrium  of  Three  Parallel  Forces.  —  Parallel  forces  are 
forces  having  parallel  lines  of  action. 

It  is  found  by  experiment  that  if  three  parallel  forces  acting 
upon  the  same  body  are  in  equilibrium,  the  following  conditions 
are  always  fulfilled  :  — 


56  Statics  of  Solids 


I 


/• 


I.     754^  three  forces ^f^^f^^  andf^  (Fig.  47),  are  in  one  plane. 

2.    The  two  outside  forces  act  in  the  same  direc- 
T-    tion  and  the  inside  force  in  the  opposite  direction, 
j[=^         3.    The  inside  force  is  equal  to  the  sum  of  the 
other  tuto, 

4.    The  outside  forces  are  ittversely  proportional 
to  the  distances  *  of  their  lines  of  action  from  the 
Fig.  47.         line  of  cu tion  of  the  inside  force  ;  that  is,  —    < 

fi  :  fi  ::  di  I  ^„  orfdi  =f^t. 

It  will  seem  that  the  inside  force  is  nearer  the  larger  of  the  other 
two ;  but  if  the  latter  are  equal,  it  is  midway  between  them.  The 
points  of  application  of  the  three  forces  need  not  lie  in  a  straight 
line.  Any  one  of  the  three  forces  may  be  regarded  as  the  equili- 
brant  of  the  other  two. 

Laboratory  Exercise  16. 

68.   Resultant  of  Two  Parallel  Forces  acting  in  the  Same  Direc- 
tion.—  When  three  parallel  forces  are  in  equilibrium, 
the  two  outside  forces  together  balance  the  third 
force;  hence  their  resultant  would  also  balance  it.    t/j 
This  resultant  {R,  Fig.  48)  must,  therefore,  have  the 
same  line  of  action  as  the  third  force,  and  must  be  ^"  ^  ' 

equal  to  it  in  magnitude  and  opposite  in  direction;  hence, — 

Tlu  resultant  of  two  parallel  forces  acting  in  the  same  direction 
is  equal  to  their  sum  ;  it  acts  in  the  same  direction,  and  its  line  of 
action  divides  the  distance  between  them  into  parts  inversely  pro- 
portional to  the  forces, 

PROBLEMS 

1.  Two  boys,  A  and  B,  carry  a  load  between  them  suspended  from  a  pole 
5  ft.  long.  The  load  is  2  ft.  from  A's  end.  What  fraction  of  it  does  A 
carry  ? 

2.  If  the  load  weighs  161  lb.,  where  must  it  be  hung  in  order  that  A  may 
carry  92  lb.  of  it  ? 

1  The  distance  between  two  lines  or  from  a  point  to  a  line  is  always  understood 
to  mean  the  perpendicular  distance. 


R 


o 


Moments  of  Force  57 


III.   Moments  of  Force 

69.  Tendency  of  a  Force  to  cause  Rotation.  —  A  force  applied 
to  a  body  may  tend  to  cause  it  to  turn  round  or  rotate  about  some 
line  as  an  axis.  The  simplest  case  is  represented  in  Fig.  49.  A 
slender  stick  (meter  rod)  is  supported  on  a 
horizontal  axis,  as  a  nail,  through  a  hole  so 
situated  that  the  rod  will  come  to  rest  in  a 
horizontal  position.  A  weight  attached  on 
either  side  of  the  axis  will   cause   rotation.  '^'  '^^' 

Two  weights,  either  equal  or  unequal,  can  be  attached,  one  on  each 
side  of  the  axis,  at  such  points  that  the  rod  will  remain  in  equi- 
librium {Exp^.  The  experiment  shows  that  the  tendency  of  a 
given  force  to  produce  rotation  is  increased  by  applying  it  farther 
from  the  axis,  and  that  the  rod  will  be  in  equilibrium  only  when 
the  product  of  one  force  and  its  distance-  from  the  axis  is  equal 
to  the  product  of  the  other  force  and  its  distance.  If/i  andy^ 
denote  the  weights,  and  ^i  and  a^  their  distances  from  the  axis 
respectively,  then  the  condition  for  equilibrium  will  be  expressed 
by  the  equation 

/i  «i  =/2  <h* 

Replacing  one  of  the  weights  by  a  drawscale,  a  measured  pull 

^  can  be  exerted  at  any  angle  (Fig.  50).     It 

^  «^\  will  then  be  found  that,  as  angle  x  increases. 


I 


a^.^  "V-*  the  force  f^  required  to  maintain  equilibrium 
1^/,  also  increases  {Exp.).  Let  a.^  now  denote 
'  the  distance  from  the  axis  to  the  ime  of  action 

'  ^  of  /a ;  then  the  condition  for  equilibrium  is 

still  /i^i  =/2«2-  The  products  /i^i  and  /iCii  therefore  measure 
the  tendencies  of  the  forces  to  cause  rotation.  The  rod  remains 
in  equilibrium  when  these  tendencies  are  equal  and  in  opposite 
directions  round  the  axis. 

70.   Moment  of  a  Force.  —  The  tendency  of  a  force  to  produce 
rotation  about  any  axis  is  measured  by  the  product  of  the  force 


58  Statics  of  Solids 

and  the  distance  of  its  line  of  action  from  that  axis.  This  product 
is  called  the  moment  of  the  force  with  respect  to  the  given  axis. 
The  distance  from  the  axis  to  the  line  of  action  of  a  force  is  called 
the  arm  of  the  force.  In  the  preceding  article  ciy^  is  the  arm  of 
the  force /i  with  respect  to  an  axis  at  O,  and  /i«i  is  the  moment 
of  the  force  about  that  axis ;  a^  is  the  arm  of  the  force /^,  and/a^^ 
its  moment  about  the  same  axis. 
Laboratory  Exercise  ij. 

71.   Equilibrium  about  an  Axis.  —  The  conditions  for  the  equi- 

ik/j  librium  of  two  forces  with  respect  to  rotation 

\  about  an  axis  are  that  their  moments  must  be 

\  equal  and  must  act   in   opposite   directions 

^\    \Zr'  '  round  the  axis.    The  condition  of  equality  is 

'^  expressed  by  the  equation  /,<z,  —fja^.     The 

t/i      Fig.  51.  forces  may  act  on  the  same  side  of  the  axis, 

as  in  Fig.  51.     The  general  law  is  as  follows  :  — 

Jn  order  that  any  number  of  forces  may  be  in  equilibrium  about 
an  axis,  the  £um  of  the  moments  of  the  forces  acting  in  one  direc- 
tion round  the  axis  must  be  equal  to  the  sum 
of  the  moments  of  the  forces  acting  in  the 
other  direction  round  it.  Thus  in  Fig.  52  the 
condition  of  equilibrium  with  respect  to  an 
axis  at  O  is/,tfi  =7,^2  ^/s^s- 

When  the  line  of  action  of  a  force  passes 
through  the  axis,  it  has  no  tendency  to  cause 
rotation,  for  the  arm  of  the  force  is  then  zero 
and  its  moment  is  zero.   This  case  is  illustrated 

byyiin  Fig.  52.     The  pressure  upon  the  body  exerted 
by  the  axis  is  also  such  a  force;   but  although  this 
^  'q    pressure  has  no  moment,  it  is  the  equilibrant  of  all 

the  other  forces  acting  upon  the  body. 

72.   Couple.  —  Two  equal  forces  acting  in  opposite 
directions  upon  the  same  body  and  having  different 
lines  of  action  constitute  a  couple  (Fig.  53).     The  dis- 
FiG.  S3.       tance  between  their  lines  of  action  is  called  the  arm 


Moments  of  Force 


59 


of  the  couple.  The  moment  of  the  couple  about  any  axis  per- 
pendicular to  the  plane  of  the  couple  is  the  product  of  its  arm 
and  either  force  (=/^  in  the  figure). 

(Prove  that  this  is  so  for  an  axis  at  O,  midway  between  the 
points  of  application  of  the  forces,  and  for  an  axis  at  Q,  a  point 
anywhere  in  the  plane.) 


PROBLEMS 

1.  If  in  Fig.  49  /i  =  90  g.,  a\  =  40  cm.,  and  a^  =  25  cm.,  what  is  the 
value  of  ^2  ? 

2.  If  in  the  same  figure  yi  =  250  g.,y^  =  125  g.,  and  a^  =  50  cm.,  what  is 
the  value  of  «i ? 

3.  How  do  two  children  of  unequal  weight  balance  each  other  in  see- 
sawing ?     Explain  how  they  make  either  end  of  the  seesaw  descend  at  will. 

4.  An  object  weighed  with  a  steelyard  (Fig.  54)  is  how  many  times  heav- 
ier than  the  weight  upon  the  beam  by  which  it  is  balanced  ? 

5.  A  man  in  moving  a  stone  with  a  crow- 
bar exerts  a  force  of  50  lb.  with  each  hand. 


I 


Fig.  54. 


Fig.  55. 


at  distances  of  100  cm.  and  150  cm.,  respectively,  from  the  point  where 
the  crowbar  is  supported,  and  this  point  of  support  is  25  cm. 
from  the  lower  end  of  the  bar.  How  great  is  the  force  exerted 
at  this  end  ? 

6.  In  drawing  a  nail  with  a  claw  hammer  a  man  exerts 
a  force  of  50  lb.  The  point  where  the  hammer  presses  against 
the  board  is  i  in.  from  the  nail  and  9  in.  from  the  point  on 
the  handle  where  the  force  is  applied.  How  great  is  the 
pull  upon  the  nail  ? 

7.  If  the  arms  of  a  balance  (Fig.  3)  are  not  of  exactly 
equal  length,  what  error  in  weighing  results  when  the  body  to  be  weighed  is 
placed  in  the  pan  on  the  longer  arm? 


Fig,  56. 


6o  Statics  of  Solids 


TV.  Effect  of  Weight  on  the  Equilibrium  of  Bodies 

73.  Gravity. — The  earth's  attraction  for  bodies  is  called  gravity^ 
when  named  in  a  general  way  without  reference  to  its  amount  for 
any  particular  body.  Gravity  acts  on  every  particle  of  a  body, 
and  these  forces  on  the  individual  particles  are  all  directed  toward 
the  same  point  at  (or  near)  the  earth's  center.  Since  this  point 
is  at  a  distance  of  about  four  thousand  miles,  there  is  no  measur- 
able error  in  assuming  that  the  forces  of  gravity  acting  on  the  parti- 
cles of  a  body  are  parallel. 

The  line  of  action  of  gravity  is  called  a  vertical  line.  It  is 
the  direction  assumed  by  a  cord  from  which  a  weight  hangs  at  rest. 
\  line  or  a  plane  perpendicular  to  a  vertical  line  is  horizontal. 

74.  Center  of  Gravity.  —  The  weight  of  a  body,  regarded  as  a 
single  force,  is  the  resultant  of  all  the  forces  of  gravity  acting  on 
its  particles.  Since  these  forces  all  act  in  the  same  direction, 
their  resultant  is  equal  to  their  sum  and  also  acts  in  the  same 
direction  —  vertically  downward.  The  point  of  application  of  the 
weight  of  a  body,  regarded  as  a  resultant  force,  is  called  the  center 
of  gravity  of  the  body.  It  is  also  called  center  of  weight,  center 
of  mass,  and  center  of  inertia. 

It  can  be  shown  both  experimentally  and  mathematically  that, 

so  long  as  a  body  retains  the  same  shape  and  distribution  of  its 

parts,  its  center  of  gravity  is  a  fixed  point 

relative  to   the   body,  however  it   may  be 

turned  and  in  whatever  condition  of  rest  or 

of  motion  it  may  be.     Further^  under  all 

circumstances  affecting  the  state  of  rest  or  of 

motion  of  a  solid,  its  weight  may  be  regarded 

as  a  single  force  applied  at  its  center  of 

gravity ;  for  the  effect  of  such  a  force  would 

be  the  same  as  that  of  the  actual  forces  of 
Fig.  57, 

gravity  acting  on  the  particles  of  the  body. 

(A  resultant  force  is  always  equivalent  to  its  components.)     For 


Equilibrium  of  Bodies 


6i 


example,  let  G  (Fig.  5  7)  represent  the  center  of  gravity  of  a  stone  ; 
then,  under  all  circumstances,  we  may  regard  its  weight  as  a 
single  force  applied  at  G  and  acting  vertically  downward. 

75.  Action  of  Weight  on  a  Suspended  Body.  —  If  a  body  is  free 
to  rotate  about  an  axis  from  which  it  is  suspended,  it  always 
comes  to  rest  with  its  center  of  gravity  vertically  below  the  axis. 
The  reason  for  this  is  shown  in  Fig.  58,  which  represents  a  flat 
body,  as  a  board,  suspended  at  O.  The  weight 
of  the  body,  w,  is  regarded  as  a  single  force 
applied  at  its  center  of  gravity,  C.  The  body 
is  evidently  not  in  equilibrium  in  the  position 
represented  in  the  figure,  since  the  moment  of 
its  weight  is  unbalanced  and  causes  rotation. 
When  C  is  vertically  below  Oy  the  moment 
of  the  weight  of  the  body  is  zero,  since  its  arm  is  then  zero.  In 
this  position  the  weight  of  the  body  and  the  pressure  upon  it  at 
the  axis  are  equal  and  opposite  and  have  the  same  line  of  action. 
This  is  therefore  a  position  of  equilibrium. 

76.  Methods  of  Finding  the  Center  of  Gravity  of  a  Body. — The 
behavior  of  the  center  of  gravity  of  a  body  when  suspended  affords 
a  simple  method  for  finding  it  experimentally,  as  illustrated  in 

Fig.  59.  The  figure  represents  any  flat  body,  as 
a  board,  suspended  at  O  upon  a  pin  or  a  nail. 
The  vertical  through  O  is  determined  by  a  small 
plumb  line  suspended  from  the  same  axis.  A  line 
indicating  the  position  of  this  vertical  is  drawn  on 
the  body.  The  center  of  gravity  of  the  body  is  at 
some  point  directly  back  of  this  line.  The  body 
is  then  suspended  at  some  other  point,  P,  and  a 
second  vertical,  DP,  determined,  as  before.     Since 

the  center  of  gravity  lies  directly  back  of  both  verticals,  it  must  be 

the  point  that  lies  directly  back  of  their  point  of  intersection  and  is 

midway  between  the  surfaces  of  the  body. 

The  center  of  gravity  of  any  body  may  be  similarly  found  by 

suspending  it  successively  at  any  two  points  about  which  it  is  free 


Fig.  59. 


62 


Statics  of  Solids 


Fig.  6o. 


to  swing  (Fig.  60).  The  only  difficulty,  if  any,  would  be  in 
accurately  determining  the  verticals  through  the 
material  of  the  body. 

The  center  of  gravity  of  a  body  of  regular  shape 
and  uniform  density  is  its  center  of  figure.  The 
centers  of  a  sphere,  a  spheroid,  an  ellipsoid,  a  cube,  a 
parallelopiped,  and  a  circular  cylinder  are  examples. 
The  shape  of  a  body  may  be  such  that  its  center  of 
gravity  does  not  lie  in  any  material  part  of  it.  This 
is  true,  for  example,  of  a  ring  or  a  hollow  sphere. 

77.  States  or  Kinds  of  Equilibrium.  —  A  body  is  in  equilibrium 
only  when  its  weight  is  balanced  by  the  other  force  or  forces  acting 
upon  it  The  equilibrium  of  a  body  in  a  given  position  is  stable  if 
the  body  tends  to  return  to  that  position  after  being  turned  or 
tilted  through  a  very  small  angle  in  any  direction ;  it  is  unstable 
if  the  body  tends  to  move  still  farther  from  its  original  position 
after  being  thus  disturbed ;  it  is  neutral  if  the  body  remains  in 
equilibrium  in  any  adjacent  position. 

Laboratory  Exercise  18. 

78.  Stable  Equilibrium.  —  A  body  suspended  at  a  point  other 
than  its  center  of  gravity  is  in  stable  equilibrium  when  at  rest  with 
its  center  of  gravity  vertically  below  the  support ;  for  the  unbalanced 
moment  of  its  weight  turns  it  back  to  this  position  when  it  is  dis- 
placed in  either  direction  (Figs.  58  and  59).  The  equilibrium  of 
a  rectangular  block  standing  on  a  horizontal  plane 

is  also  stable.  When  slightly  displaced  by  tilting, 
the  moment  of  its  weight  about  the  edge  on  which 
it  is  turned  tends  to  bring  it  back  to  its  former 
position  (Fig.  61). 

When  any  body  in  stable  equilibrium  is  inclined, 
its  center  of  gravity  is  raised  ;  and  in  returning  to 
the  position  of  equilibrium,  the  center  of  gravity 
falls.  Every  body  at  rest  is  in  stable  equilibrium 
when  its  position  is  such  that  the  slightest  inclination  in  any  direc- 
tion raises  its  center  of  gravity. 


c\ 


w 
Fig.  61. 


Equilibrium  of  Bodies  63 

79.  Unstable  Equilibrium.  —  A  body  is  in  unstable  equilibrium 
when  balanced  on  a  point,  an  edge,  or  an  axis  with  its  center  of 
gravity  vertically  above  the  support.  Figure  62  repre- 
sents a  rectangular  block  balanced  in  unstable  equi- 
librium on  an  edge.  In  this  position  of  the  block  the 
moment  of  its  weight  is  zero ;  but,  with  the  slightest 
displacement  in  either  direction,  the  moment  of  its 
weight  tends  to  turn  the  block  still  farther  in  the  same 
direction.  An  egg  standing  on  end  upon  a  flat  sur- 
face would  be  in  unstable  equilibrium.  The  center  of  gravity  of  a 
body  in  unstable  equilibrium  is  higher  than  it  would  be  in  adjacent 
positions  of  the  body ;  hence  the  slightest  disturbance  causes  such 
a  body  to  fall. 

80.  Neutral  Equilibrium.  —  A  body  supported  on  an  axis  pass- 
ing through  its  center  of  gravity  is  in  neutral  equilibrium  when  at 
rest  in  any  position,  since  the  moment  of  its  weight  is  zero  for 
all  positions  of  the  body  about  the  axis.  When  a  body  thus  sup- 
ported is  set  rotating,  it  will  come  to  rest  in  any  position  in  which 
friction  may  chance  to  stop  it.  A  perfectly  balanced  wheel  is  a 
familiar  illustration  of  neutral  equilibrium  ;  but  any  body,  however 
irregular  in  shape,  may  be  suspended  in  neutral  equilibrium.  The 
equilibrium  of  a  perfect  sphere  at  rest  on  a  horizontal  plane  is 
neutral,  since  the  vertical  through  its  center  of  gravity  passes 
through  the  point  of  support  in  all  positions.  The  center  of 
gravity  of  a  body  is  neither  raised  nor  lowered  by  any  slight  dis- 
placement from  a  position  of  neutral  equilibrium. 

A  body  may  be  in  different  states  of  equilibrium  at  the  same 
time  with  respect  to  motion  in  different  directions.  Thus  a  cir- 
cular cylinder  at  rest  on  its  side  upon  a  horizontal  surface  is  in 
neutral  equilibrium  with  respect  to  rolling,  and  in  stable  equilibrium 
with  respect  to  motion  by  which  either  end  would  be  raised. 

81.  An  Important  Property  of  the  Center  of  Gravity.  —  Suppose 
a  body  of  irregular  shape  (Fig.  63)  to  be  suspended  in  neutral 
equilibrium  at  its  center  of  gravity,  C.  Let  c  be  the  center  of 
gravity,  and  w  the  weight  of  the  portion  of  the  body  lying  on  one 


64 


Statics  of  Solids 


side  of  a  vertical  plane  through  C,  and  c'  and  w'  the  center  of 

gravity  and  weight  respectively  of 
the  portion  lying  on  the  other  side 
of  this  plane.  Also  let  a  and  a^ 
be  the  arms  of  w  and  7v'  respec- 
^*  tively  about  C  as  an   axis.     Then, 

since  the  body  is  in  equilibrium,  wa  =  7v^a'.  This  stated  in  words 
means  that  a  plane  through  the  center  of  gravity  of  a  body  divides 
the  body  into  two  parts  whose  weights  have  equal  moments  about 
the  center  of  gravity. 

For  irregular  bodies  the  arms  a  and  a'  will,  in  general,  be 
unequal  ;  in  which  case  10  and  7v'  will  also  be  unecjual.  Hence  a 
plane  through  the  center  of  gravity  of  a  body  generally  divides  it 
into  parts  having  unequal  weight. 

82.  Stability.  —  It  is  well  known  that  a  body  is  overturned 
with  greater  difficulty  when  the  area  of  the  base  upon  which  it 


stands  is  increased.  A  rect- 
angular block  of  wood  provided 
with  a  large  base  at  one  end 
affords  excellent  illustration 
(Fig.  64).  It  is  in  stable  equi- 
librium in  either  position  shown 
in  the  figure  ;  but  its  stability 
is  much  greater  when  standing 
upon  the  larger  base.    The  rea- 


4:71 


r:d" 


ic 


7 


C 


Fig.  64. 


son  for  this  is  that  in  the  latter  position  the  moment  of  the  weight 
of  the  body,  wa,  is  much  greater,  and  also  that 
the  body  must  be  turned  much  farther  against 
this  moment  before  it  will  fall  over. 

The  stability  of  a  body  standing  on  a  given 
base  is  increased  by  lowering  its  center  of 
gravity.  This  may  be  illustrated  by  a  block 
weighted  with  lead  at  one  end.  When  stand- 
ing on  its  heavier  end,  the  block  must  be  turned 

farther  before  its  weight  acts  to  overturn  it  (Fig.  65). 


Fig.  65. 


Equilibrium  of  Bodies  65 

The  two  conditions  for  great  stability  are,  therefore,  a  large  base 
and  a  low  ce?iter  of  grain ty.  So  far  as  stability  is  concerned,  the 
base  of  a  body  standing  on  legs  is  the  entire  area  inclosed  within 
straight  lines  connecting  the  legs. 

Laboratory  Exercise  ig. 

83.  Equilibrium  of  Floating  Bodies.  —  The  buoyant  force  upon 
a  floating  body  is  the  resultant  of  the  total  pressure  of  the  liquid 
upon  it.  This  resultant  is  equal  to  the  weight  of  the  body  (Art.  32), 
its  direction  is  always  vertically  upward,  and  its  point  of  applica- 
tion is  the  center  of  gravity  of  the  displaced  liquid.  This  point 
is  called  the  center  of  buoyancy.  We  may  therefore  regard  a  float- 
ing body  as  being  acted  upon  by  two  resultant  forces  —  weight 
and  buoyancy  —  which  are  always  equal  and  opposite.  A  float- 
ing body  is  in  equilibrium  when  these  forces  have  the  same  line 
of  action  ;  that  is,  when  the  center  of  gravity  of  the  body  and  the 
center  of  buoyancy  lie  in  the  same  vertical  line.  This  is  illus- 
trated by  the  first  and 

third  parts  of  Fjg.  66,  1 

in   which    C  denotes     ^!^^      \b 


the  center  of  gravity     ^=-3 ^•:gj^>5^^^^^fej  t^  F^^:=d 

of  the  body  and  ^the  ~^"         ~ 

/  ,  Fig.  66. 

center    of    buoyancy. 

When  a  floating  body  is  displaced  from  its  position  of  equilibrium, 
the  lines  of  action  of  weight  and  buoyancy  no  longer  coincide,  and 
the  two  forces  constitute  a  couple  whose  effect  is  to  restore  equi- 
librium, if  the  equilibrium  was  stable,  or  to  carry  the  body  still 
farther  from  that  position,  if  the  equilibrium  was  unstable.  When 
the  rectangular  block  represented  in  the  figure  is  tilted  from  its 
first  position,  the  center  of  buoyancy  shifts  toward  the  deeper 
displacement,  while  the  position  of  the  center  of  gravity  remains 
unchanged.  This  establishes  a  couple  which  tends  to  restore  the 
body  to  its  former  position.  Equilibrium  in  the  first  position  is 
therefore  stable.  The  third  part  of  the  figure  represents  the  posi- 
tion of  unstable  equilibrium  {Exp.).  (Illustrate  this  in  a  figure 
showing  the  body  slightly  displaced.) 

i 


66  Statics  of  Solids 

The  equilibrium  of  a  floating  body  is  always  stable  when  the 
center  of  gravity  of  the  body  is  below  the  center  of  buoyancy ; 
when  it  is  above  the  center  of  buoyancy,  the  equilibrium  may  be 
either  stable  or  unstable,  depending  upon  the  shape  and  position 
of  the  body,  as  illustrated  above.  The  equilibrium  is  neutral  if 
the  centers  of  gravity  and  buoyancy  remain  in  the  same  vertical 
line  when  the  l)ody  is  disturbed,  as  is  the  case  with  a  sphere  or  a 
long  cylinder  with  its  axis  horizontal  {Exp.), 

84.  Stability  of  Floating  Bodies. —The  stability  of  a  floating 
body  of  given  shape  is  increased  by  lowering  its  center  of  gravity ; 
for  this  increases  the  arm  of  the  couple,  which  tends  to  right  the 
body  when  displaced.  It  is  for  this  reason  that  a  vessel  witho'ut 
a  cargo  carries  ballast. 

PROBLEMS 

I.  In  what  direction  does  a  person  lean  when  carrying  a  heavy  load 
in  one  hand  ?    Why  ? 

2.  Show  that  when  a  homogeneous  hemisphere  is 
inclined  {A^  Fig.  67),  its  weight  tends  to  bring  it  into 
the  position  shown  in  B.  In  what  kind  of  e(|uilibrium 
is  it  in  the  second  position  ?  Is  it  in  unstable  equi- 
librium in  the  first  position  ?     Give  reasons. 

3.  (a)  Oil  cans  are  made  of  the  shape  shown  in  Fig.  68,  and  are 
weighted  with  lead  at  the  bottom.  Such  a  can  rights  itself  when 
tipped.  Explain.  (^)  Does  the  can  really  rise  or  fall  when  it 
rights  itself?  fiG.  68. 

4.  Why  does  a  person  always  lean  forward  before  attempting  to  rise  from 
a  chair  ? 

5.  A  pencil  with  a  knife  attached  can  be  balanced,  as  shown  in  Fig.  69. 
Try  it.  ^^^lat  is  the  evidence  that  the  equilibrium  is  stable  ? 
Where  is  the  center  of  gravity  of  the  pencil  and  knife  regarded  as 
one  body  ? 

6.  Show  by  means  of  figures  that  the  .moment  of  the  weight  of 
a  sphere  is  zero  upon  a  horizontal  surface,  but  not  upon  an  inclined 
plane. 

7.  If  a  body  that  will  not  roll  remains  at  rest  when  placed  on 
an  inclined  plane,  three  forces  act  to  hold  it  in  equilibrium.     Two 

Fig.  69.       of  these  forces  are  its  weight  and  the  pressure  of  the  plane.     What 


Equilibrium  of  Bodies  67 

is  the  third  force,  and  in  what  direction  does  it  act  ?     Draw  a  figure  correctly 
representing  the  direction  and  the  relative  magnitude  of  the  three  forces. 

8.  Two  spheres  weighing  50  kg.  and  15  kg.,  respectively,  are  connected 
by  a  rod  so  that  the  distance  between  their  centers  is  80  cm.  Disregarding 
the  weight  of  the  rod,  where  is  the  center  uf  gravity  of  the  whole  considered 
as  one  liody  ? 

9.  The  average  distance  between  the  centers  of  the  earth  and  the  moon 
is  about  240,000  miles  ;  the  mass  of  the  earth  is  80  times  that  of  the  moon. 
How  far  is  their  common  center  of  gravity  from  the  earth's  center? 

10.  Two  men,  A  and  B,  carry  a  board  30  ft.  long  and  of  uniform  cross- 
section.  A  holds  at  one  end  ;  where  must  B  hold  in  order  to  carry  .6  of  the 
load? 

11.  A  boy  weighing  40  lb.  wishes  to  seesaw  alone  on  a  plank  weighing 
70  lb.  The  plank  is  24  ft.  long,  and  the  center  of  gravity  of  the  boy  is  i  ft. 
from  an  end  of  the  plank.  How  far  from  that  end  must  the  plank  be  sup- 
ported ? 

12.  Why  is  it  an  advantage  to  spread  the  feet  when  standing  upon  a  sur- 
face that  is  moving  unsteadily,  as  the  deck  of  a  vessel  ? 

13.  What  would  happen  to  the  leaning  tower  of  Pisa  (Fig.  75)  if  the 
vertical  through  its  center  of  gravity  fell  without  the  base  of  the  tower  ? 

14.  Is  the  stability  of  a  boat  greater  when  the  occupants  are  standing  or 
sitting  ?     Why  ? 

15.  Why  is  it  difficult  to  walk  on  stilts  ? 

16.  A  uniform  stick  oC  timber  10  ft.  long  balances  on  an  axis  3  ft.  from 
one  end  when  a  weight  of  20  lb.  is  hung  from  that  end.  Find  the  weight  of 
the  stick. 

17.  Why  cannot  one  stand  with  hb  heels  against  a  wall  and  lean  forward 
without  falling  ? 

18.  Two  boys,  A  and  B,  carry  a  uniform  plank  24  ft.  long,  weighing  120 
lb.  A  holds  at  one  end  and  B  4  ft.  from  the  other  end.  What  load  does 
each  carry  ? 


CHAPTER  V 

DYNAMICS 

I.  Motion 

85.  Motion. —  Motion  is  continuous  change  of  position.  The 
line  along  which  the  center  of  gravity  of  a  body  moves  is  regarded 
as  the  path  of  the  body.  The  motion  of  a  body  is  completely 
known  when  we  know  its  path  and  the  rate  of  motion  at  every 
point  of  the  path,  or  the  rate  and  direction  of  motion  at  every 
instant  during  the  motion. 

Rate  of  motion  is  called  speed.  Velocity  includes  both  rate  and 
direction  of  motion.  The  distinction  in  meaning  between  the  two 
words  is  frequently  useful,  but  it  is  not  strictly  adhered  to.  Thus 
the  word  velocity  is  frequently  used  to  signify  merely  rate  of 
motion,  its  direction  not  being  a  matter  of  importance  for  the 
purpose  under  consideration. 

86.  Uniform  Motion.  —  The  motion  of  a  body  is  uniform  if  the 
body  passes  over  equal  portions  of  its  path  in  equal  intervals  of 
time,  however  short  these  intervals  may  be.  If  the  motion  of  a 
body  is  imiform,  its  speed  is  constant,  and  is  measured  by  the 
distance  that  the  body  moves  over  in  a  unit  of  time. 

Speed  is  measured  in  various  combinations  of  units  of  distance 
and  of  time,  as  centimeters  per  second,  meters  per  second,  feet 
per  second,  miles  per  hour,  etc.  In  the  definitions  that  follow, 
the  second  will  be  named  as  the  unit  of  time,  since  it  is  the  only 
one  that  is  used  in  scientific  work. 

The  whole  distance  passed  over  by  a  body  moving  with  constant 
speed  is  equal  to  the  product  of  the  speed  and  the  time  occupied 
in  traversing  the  distance.     Hence,  letting  d  denote  the  distance, 

68 


Motion  69 

V  the  (magnitude  of  the)  velocity,  and  /  the  time,  we  have  for 
uniform  motion :  — 

d—vt'.  also  V  =  -,  and  /=  -•  (i) 

87.  Variable  Motion. — The  expression  "the  velocity  of  a 
body  "  has  no  definite  meaning  unless  the  velocity  is  constant.  If 
it  is  variable,  qualifying  terms  are  required,  as  indicated  in  the 
following  definitions :  — 

T/ie  velocity  of  a  body  at  any  instant  (or  at  any  point  of  its  path) 
is  the  distance  that  it  would  pass  over  during  the  next  second  (or 
other  unit  of  time)  if  its  velocity  continued  unchanged  from  that 
instant.  Thus  when  we  say  that  a  train  is  running  at  the  rate  of 
30  miles  per  hour,  we  mean  that  it  would  run  30  miles  in  an  hour 
if  it  continued  at  its  present  rate  for  one  hour. 

The  average  velocity  of  a  body  during  any  interval  of  time  (or 
between  any  two  points  of  its  path)  is  the  uniform  velocity  that 
would  be  required  to  pass  over  the  same  distance  in  the  same 
time.  Average  velocity  is  therefore  equal  to  the  distance  divided 
by  the  time.  Representing  average  velocity  by  z>,  its  definition  is 
expressed  by  the  formula, 

d—vt)  from  which  v  =  -*  (2) 

For  example,  if  an  automobile  runs  108  mi.  in  6  hr.,  its 
average  rate  is  18  mi.  per  hr.,  since  this  is  the  uniform  rate 
required  to  run  the  given  distance  in  the  given  time.  The  actual 
rate  may  vary  from  o  (during  intervals  of  stopping)  to  40  or  50 
mi.  per  hr. 

88.  Representation  of  Velocities. — A  velocity  may  be  repre- 
sented in  both  magnitude  and  direction  by  a    b.^ 
straight  line,  just  as  a  force  may  be.     Thus  if 
OA  (Fig.  70)  represents  a  velocity  of  3  ft.  per 

sec.  east,  then  OB  represents  a  velocity  of  2  ft.     o' ■ ^^ 

per  sec.  north.  ^^'''  7°- 

89.  Composition  of  Velocities.  — A  body  may  have  two  or  more 
independent   motions  at  the  same  tinae    {Exp.).      Thus  a   boat 


70 


Dynamics 


rowed  across  a  stream  has  a  motion  imparted  by  the  rowing,  and 
also  a  motion  due  to  the  current  and  equal  to  it.  Suppose  the 
boat  to  be  constantly  headed  directly  toward  the  opposite  shore, 
and  let  O  (Fig.  71)  represent  the  starting  point. 
OB  would  be  the  path  of  the  boat  if  there  were 
no  current.  OC  is  the  distance  the  stream  flows 
while  the  boat  is  crossing.  The  actual  motion 
of  the  boat  is  the  resultant  of  these  two  motions ; 
its  path  is  represented  by  OA,  If  OB  and  OC 
be  taken  to  represent  the  component  ve/ocitieSt 
then  OAf  the  concurrent  diagonal  of  the  paral- 
lelogram constructed  on  OB  and  OC  as  sides, 
will  represent  the  actual,  or  resultant,  velocity  upon  the  same  scale. 
Velocities  are,  in  fact,  compounded  by  the  same  rules  as  forces 
(.\rts.  63  and  64).  The  construction  is  called  the  parallelogram 
of  velocities. 

90.  Resolution  of  a  Velocity.  —  A  velocity,  like  a  force,  can 
be  resolved  into  components  in  any  chosen  directions.  The  con- 
struction is  the  same  as  for  the  resolution  of  a  force  (Art.  66). 
For  example,  a  vessel  is  sailing  30°  north  of  east  at  the  rate  of  1 2 


Fig.  71. 


mi.  per  hr.  At  what  rate  is  it  advancing  north- 
ward and  at  what  rate  eastward  ?  It  is  proved  in 
geometry  that  in  a  right  triangle  having  an  acute 
angle  of  30°  the  hypothenuse  is  twice  the  shorter  ^^^-  7a- 

leg.     Hence    ON  (Fig.    72),    the   northerly   component   of  the 
velocity,  is  6  mi.  per  hr. ;  and  OE,  the  easterly  component,  is 


Fig.  73. 


V12*  —  6*  =  10.4-  mi.  per  hr. 

As  a  further  illustration,  let  us  consider  how  the 
boat  mentioned  in  the  preceding  article  must  be 
rowed  in  order  to  reach  the  opposite  bank  at  B 
instead  of  at  A.  The  resultant  motion  is  now  rep- 
resented by  OB,  The  component  OC,  due  to  the 
motion  of  the  stream,  is  the  same  as  before.  Hence 
OB  is  the  diagonal  of  a  parallelogram  of  which 


Motion  71 

one  side  is  OC.  The  other  component  motion  is  therefore  repre- 
sented by  OA^  (Fig.  73).  This  means  that  the  boat  must  be  con- 
stantly pointed  in  a  direction  parallel  to  0A\  and  that  it  would 
take  as  long  to  reach  B  as  it  would  to  row  the  distance  OA'  in 
still  water.  (In  the  preceding  problem  of  Art.  89  would  more 
time  be  required  to  cross  to  A  than  to  cross  to  B  in  still  water  ?) 

PROBLEMS 

1.  A  ball  rolls  53  m.  in  ii  sec.     Find  its  average  velocity. 

2.  A  train  runs  with  an  average  velocity  of  23  m.  per  sec.  In  what  time 
does  it  run  a  kilometer  ? 

3.  From  a  train  running  at  the  rate  of  9  m.  per  sec,  a  mail  bag  is  thrown 
at  right  angles  to  the  track  with  a  velocity  of  4  m.  per  sec.  Compute  the 
resultant  velocity  of  the  bag  at  the  instant  it  leaves  the  hand,  and  draw  a  fig- 
ure to  show  its  direction. 

4.  From  a  train  running  at  the  rate  of  12  m.  per  sec.  a  mail  bag  is  thrown 
so  that  its  resultant  velocity  is  equal  to  that  of  the  train  and  at  right  angles 
to  it.  What  is  the  magnitude  and  direction  of  the  velocity  imparted  in 
throwing  the  bag  ? 

5.  An  arrow  is  shot  directly  backward  from  the  rear  of  a  train  with  a 
velocity  (relative  to  the  train)  equal  to  that  of  the  train.  What  is  the  motion 
of  the  arrow  ? 

6.  The  rotation  of  the  earth  carries  its  surface  eastward  at  the  rate  of 
about  \  mi.  per  sec.  (in  temperate  latitudes).     When  a  ball  is 
thrown  up,  why  is  it  not  left  behind  (to  the  west)  by  the  earth 
in  its  rotation  ? 

7.  Four  boys,  A,  B,  C,  and  D  (Fig.  74),  on  the  deck  of  a 
moving  vessel,  pass  a  ball  round  in  the  order  of  the  letters. 
What  allowance  for  the  motion  of  the  vessel,  if  any,  must  be 
made  by  each  of  the  boys  in  throwing  ?     Give  reasons.  '  '^ 

8.  A  vessel  sails  due  N.E.  at  the  rate  of  15  mi.  per  hr.  Compute  the 
northerly  and  easterly  components  of  its  velocity. 

9.  A  boat  is  rowed  so  that  it  crosses  a  stream  100  m.  wide  to  a  point  directly 
opposite  to  the  starting  point  (Fig.  73).  The  stream  flows  .8  m.  per  sec, 
and  the  boat  is  rowed  at  the  rate  of  1.2  m.  per  sec.  in  still  water.  How  long 
is  the  boat  crossing  the  stream  ? 

91.  Acceleration.  —  The  velocity  of  a  body  is  said  to  be  accel- 
erated when  it  is  increasing,  and  retardedy  or  ?iegative/y  accelerated, 
when  it  is  decreasing.     The  rate  of  change  of  velocity  is  called  the 


72  Dynamics 

acceUration.  Thus,  if  a  body  starting  from  a  state  of  rest  has  a 
velocity  of  3  m.  per  sec.  at  end  of  the  first  second,  6  m.  per  sec. 
at  the  end  of  the  second  second,  9  m.  per  sec.  at  the  end  of  the 
third  second,  etc.,  its  velocity  increases  3  m.  per  sec.  every  second  ; 
i.e,  its  acceleration  is  3  m.  per  sec.  per  sec.  This  is  a  case  of 
uniformly  atceUraUd  motion^  or  constant  acceleration^  the  increase 
of  velocity  being  the  same  for  each  second.  When  the  velocity 
of  a  body  decreases  by  the  same  amount  during  each  second, 
its  motion  is  said  to  be  uniformly  retarded^  or  to  have  a  constant 
negative  acceleration. 

It  more  frequently  happens  that  the  acceleration  of  a  body  is 
variable.  This  is  the  case,  for  example,  with  a  street  car.  As 
its  speed  increases,  the  rate  of  increase  diminishes.  When  its 
speed  is  the  greatest,  the  increase  of  speed,  or  acceleration,  is  zero. 
There  are,  however,  important  cases  of  constant  acceleration  ;  and 
it  is  only  these  that  we  shall  consider  quantitatively. 

Motion  is  accelerated  when  it  changes  in  direction,  even  if  the 
speed  remains  constant.  This,  however,  is  reserved  for  later  con- 
sideration (Art.  119). 

92.  Formulas  for  Uniformly  Accelerated  Motion.  —  In  the 
case  of  uniformly  accelerated  motion  in  a  straight  line,  the  accel- 
eration is  measured  by  the  constant  change  of  speed  that  occurs 
during  each  second.  If  a  body  moves  with  a  constant  accelera- 
tion of  a  cm.  per  sec.  per  sec,  starting  from  rest,  its  velocity  at  the 
end  of  I  sec.  will  be  a  (centimeters  per  sec),  at  the  end  of  2  sec 
it  will  be  2  /7,  at  the  end  of  /  sec.  it  will  be  /  a.    This  is  expressed 

by  the  formula 

v^at,  (3) 

in  which  a  denotes  the  constant  acceleration  and  v  the  velocity  at 
the  end  of  /  sec.  after  starting.  This  is  usually  called  the  final 
velocity. 

Since  the  velocity  increases  from  zero  at  a  uniform  rate,  the 
average  velocity,  z>,  during  the  time  /,  is  one  half  of  the  final  veloc- 
ity ;  t^.v=—.    The  entire  distance  traversed  by  the  body  is  the 


Motion  73 

product  of  its  average  velocity  and  the  time  (Art.  87)  ;  hence, 
letting  d  denote  this  distance, 

.      -,     at      ^     at^ 
d=vt  =  —Xt  =  —  ; 
2  2 

.0  /    ^ 

that  is,  d= — ;  and  /=\/ (4) 

V 

From  equation  (3),  /=-.     Substituting  this  value  of  /  in  equa- 
tion (4),  we  have 


2  2  a 


zr' 


that  is,  d= — -,  and  v^y/tad,  (5) 


2a 


This  formula  expresses  the  relation  between  the  constant  accel- 
eration of  a  body  starting  from  rest,  the  distance  that  the  body  has 
traversed,  and  its  velocity  at  the  end  of  that  distance. 

If  any  two  of  the  quantities  in  formula  (3),  (4),  or  (5)  are  given, 
the  value  of  the  third  quantity  can  be  found  by  substituting  the 
given  values  in  the  formula. 

93.  Laws  of  Uniformly  Accelerated  Motion.  —  The  following 
laws  of  uniformly  accelerated  motion  in  a  straight  line,  for  bodies 
starting  from  rest,  are  contained  in  the  above  formulas :  — 

I.  The  velocity  at  any  instant  is  proportional  to  the  time  during 
which  the  body  has  been  in  motion.     (Formula  3.) 

II.  The  velocity  acquired  in  a  given  time  is  proportional  to  the 
acceleration.     (Formula  3.) 

III.  The  average  velocity  during  the  whole  time  is  half  the  final 
velocity. 

IV.  The  distance  passed  over  is  proportional  to  the  square  of 
the  time.  For,  if  the  body  traverses  the  distance  d^  in  t^  sec.  and 
the  distance  d.,  in  /o  sec,  both  measured  from  the  instant  of  start- 
ing, then,  from  formula  (4),  /?i  =  — ^,  and  4=  — ^.     Dividing  each 

2  2 


74  Dynamics 

member  of  the  first  equation  by  the  corresponding  member  of  the 
second,  we  get    .  =  tj  »  hence, 

d,'d^.'.  A*  :  i,\  (6) 

V.  Thf  distance  trm^ersed  in  a  given  time  is  proportional  to  the 
acceleration.     (Formula  4.) 

VI.  The  acceleration  is  numerically  equal  to  twice  the  distance 
trax>ersed  during  the  first  second. 

For,  when  /=  i,  formula  (4)  becomes  d=  -. 

2 

PROBLEMS 

1.  A  ftreet  car  rant  with  a  constant  acceleration  of  1.2  m.  per  sec.  per  sec. 
for  8  sec.  after  starting,  (a)  What  is  its  velocity  at  the  end  uf  that  time? 
(^)  What  was  its  average  velocity  during  the  8  sec?  (r)  How  far  does  it 
ran  in  the  8  sec.? 

2.  A  stone  falls  with  a  constant  acceleration  of  980  cm.  per  sec.  per  sec. 
In  what  time  will  it  acquire  a  velocity  uf  35  m.  per  sec.  ? 

3.  A  body  moves  with  a  constant  acceleration  a.  (a)  How  far  does  it  go 
in  the  first  second  ?  (^)  What  is  its  average  velocity  during  the  first  second? 
(r)  What  is  the  average  velocity  during  the  first  6  sec?  (</)  What  is  the 
average  velocity  during  the  sixth  sectmd? 

4.  A  train,  running  with  constant  acceleration,  goes  560  m.  during  the 
first  minute  after  starting.     Find  the  acceleration  in  meters  per  sec.  per  sec. 

5.  A  car  runs  with  a  constant  acceleration  uf  80  cm.  per  sec.  per  sec.  for 
a  distance  of  300  m.  (<i)  What  is,  then,  its  velocity  ?  (^)  With  what  average 
velocity  did  it  ran  that  distance?  {c)  How  long  did  it  take  to  run  this  dis- 
tance? 

6.  A  ball  rolling  along  the  ground  is  uniformly  retarded  at  the  rate  of 
4  m.  per  sec.  per  sec.  Its  velocity  at  the  start  is  20  m.  per  sec.  (a)  How 
long  will  it  roll  ?     (^)  How  far  will  it  roll  ? 

n.  Falling  Bodies 

94.  Relative  Rate  of  Falling  Bodies  ;  Resistance  of  the  Air.  — 
An  unsupported  body  falls  because  the  attraction  of  the  earth  upon 
it  (i>.  the  weight  of  the  body)  is  unbalanced.  It  is  well  known 
that  a  feather  or  a  sheet  of  paper  falls  less  rapidly  than  a  stone, 
and  that  some  bodies  —  a  balloon,  for  example  —  rise  instead  of 


Falling  Bodies  75 

falling.  From  such  familiar  facts  as  these  false  conclusions  are  fre- 
quently drawn ;  but  the  truth  can  be  gathered  from  a  careful  study 
of  a  few  simple  experiments. 

We  know  that  a  balloon  rises  because  its  weight  is  less  than  the 
buoyancy  of  the  air,  leaving  an  unbalanced  force  acting  upward. 
Since,  however,  the  buoyant  force  of  the  air  upon  solids  and 
liquids  is  relatively  very  small  (Art.  55),  it  cannot  appreciably 
affect  their  rate  of  fall. 

If  we  take  two  sheets  of  paper  exactly  alike  and  roll  one  of 
them  into  a  tight  wad,  it  will  be  found,  on  dropping  them  simul- 
taneously from  the  same  height,  that  the  wad  falls  much  faster 
than  the  open  sheet  {Exp.).  The  difference  in  their  rate  of  fall 
is  due  to  the  friction  of  the  air,  which  is  greater  upon  the  open 
sheet,  since  it  has  the  greater  surface  exposed.  Buoyancy  is  evi- 
dently not  the  cause  of  the  difference,  since  the  buoyant  force 
upon  the  sheet  is  the  same  whatever  its  shape. 

If  two  pebbles  of  very  unequal  size  are  held,  one  in  each  hand, 
above  the  head  at  the  same  height,  and  dropped  at  the  same  in- 
stant, they  will  reach  the  ground  together.  (Try  it.)  In  this 
experiment  the  friction  of  the  air  is  too  small  to  produce  an 
appreciable  effect  upon  either  body  ;  hence  their  observed  motion 
may  be  regarded  as  due  to  their  weight  alone.  This  experiment 
illustrates  the  interesting  fact  that  all  bodies  not  appreciably 
affected  by  the  resistance  of  the  air  fall  at  the  same  rate,  regard- 
less of  their  weight.  The  pupil  should  test  this  further  by  compar- 
ing in  pairs  the  rates  of  fall  of  a  number  of  different  bodies. 

Actual  differences  in  the  observed  rates  of  fall  may  be  assumed 
to  be  due  to  friction  of  the  air ;  and  the  pupil  should  discover  by 
experiment  the  approximate  effect  of  friction  upon  bodies  of 
widely  different  density,  by  trying  simultaneously  stone,  wood, 
cork,  wad  of  paper,  etc.  The  result  of  greater  or  less  compact- 
ness of  form  can  be  observed  by  dropping  together  a  wad  of 
paper  and  an  open  sheet,  a  block  and  a  very  thin  board  or  a  leaf, 
etc.  The  effect  of  the  air  for  greater  velocities  is  easily  tested  by 
dropping  the  bodies  out  of  a  second  or  third  story  window. 


76  Dynamics 

95.  HistoricaL  —  Experiments  similar  to  the  above  were  first 
tried,  so  far  as  is  known,  by  Galileo  Galilei  (i 564-1 642),  an 
Italian  mathematician  and  scientist.  For  two  thousand  years  no 
one  had  thought  to  question  the  doctrine  of  the  Greek  philoso- 
pher Aristotle,  who  taught  that  the  rate  of  fall  of  bodies  was  pro- 
portional to  their  weight.  Galileo,  who  more  fully  appreciated 
the  value  of  experiment  than  any  of  his  predecessors,  discovered 

^B^  the  falsity  of  this  doctrine,  and  proved  the 

J^^L  correctness  of  his  view  to  the  citizens  of  Pisa 

H^^B  by  dropping  simultaneously  a  one-pound  ball 

^^^^t  and  a  one-hundred-pound  ball  from  tlie  top 

^^^^m  of  the   leaning   tower  (P'ig.  75).     The   two 

^^^^H  balls  reached  the  ground  together. 

-^^^^K  l'^^^  Any  difference  in  the  rate  of  fall  of 

.'^'^^^BHB^         bodies  is  due  to  the  resistance  of  the  air  was 

^^        clearly  proved,  after  the  invention  of  the  air 

pump,  by  causing  such  bodies  as  a  coin  and  a 

feather  to  fall  together  from  one  end  to  the 

other  of  a  long  glass  tube  from  which  the  air 

F»«-  7S«  had  been  exhausted.     The  feather  was  found 

to  fall  as  rapidly  as  the  coin.     The  experiment  is  known  to  the 

present  day  as  the  guinea  and  feather  experiment 

Laboratory  Exercise  2t. 

96.  Acceleration  of  Falling  Bodies.  —  Our  daily  experience 
teaches  that  the  speed  of  bodies  continually  increases  as  they  fall ; 
/>.  bodies  fall  with  accelerated  motion.  Since  all  bodies  fall 
equal  distances  in  equal  times,  unless  appreciably  affected  by  the 
resistance  of  the  air,  it  is  evident  that  gravity,  acting  alone,  ac- 
celerates all  bodies  equally.  It  can  be  shown  by  a  number  of 
experimental  methods  that  this  acceleration  is  uniform. 

One  of  the  simplest  and  most  accurate  methods  by  which  the 
motion  of  a  falling  body  can  be  studied  is  that  known  as  Whit- 
ing's pendulum  method  (Lab.  Ex.  21).  In  this  experiment  the 
distances  </,  and  4  through  which  a  ball  falls  in  measured  inter- 
vals of  time  /i  and  /,  are  determined,  the  ball  in  each  case  starting 


Falling   Bodies  77 

from  a  state  of  rest.  The  values  obtained  will  agree  (within  the 
limits  of  experimental  error)  with  the  relation 

d,'.d,::t^'.t!',  (6) 

that  is,  the  distance  that  a  body  falls  from  a  state  of  rest  is  pro- 
portional to  the  square  of  the  time.  Since  this  is  one  of  the  laws 
of  uniformly  accelerated  motion  (Law  IV,  Art.  93),  it  proves  that 
the  acceleration  of  a  falling  body  is  uniform. 

97.  Laws  of  Falling  Bodies.  —  All  the  formulas  and  laws  for 
uniformly  accelerated  motion  (Arts.  92  and  93)  hold  for  falling 
bodies,  their  motion  being  uniformly  accelerated  ;  but  in  this  case 
the  acceleration  is  denoted  by  g^  since  it  is  due  to  gravity.  The 
motion  of  a  body  falling  freely  from  a  state  of  rest  (and  hence 
falling  vertically)  is,  therefore,  completely  expressed  by  the 
formulas:-  ^^^,.  ^^^ 

^=f;  (8) 

^=?;  (9) 

-.=  il.  (:o) 

98.  The  Value  of  ^.  —  Since  the  weight  of  a  body  changes 
slightly  with  a  change  of  latitude  or  of  altitude,  the  acceleration 
that  it  causes  also  varies.  Thus  at  the  equator,  where  the  weight 
of  a  body  is  least,  the  value  of^  is  978  cm.  per  sec.  per  sec,  while 
at  the  poles,  where  weight  is  greatest,  it  is  983  cm.  For  places 
within  the  temperate  zones  980  cm.  or  32.15  ft.  per  sec.  per  sec. 
is  a  very  close  approximation ;  and  this  value  is  to  be  used  in 
solving  problems. 

99.  Cause  of  the  Constant  Acceleration  of  Falling  Bodies. — 
The  weight  of  a  body  is  a  constant  force,  which,  so  long  as  the 
friction  of  the  air  is  inappreciable,  is  wholly  unbalanced  during 
free  fall.  The  constant  acceleration  of  a  falling  body  is  therefore 
due  to  the  continuous  action  of  a  constant  unbalanced  force. 


78  Dynamics 

100.  Effect  of  Friction  of  the  Air.  —  The  resistance  to  motion 
due  to  the  friction  of  the  air  increases  rapidly  with  the  velocity. 
Bicycle  riders  are  familiar  with  an  excellent  illustration  of  this. 
For  example,  to  ride  on  a  level  road  at  the  rate  of  10  mi.  an  hr. 
in  the  direction  of  a  wind  of  equal  velocity  requires  but  little 
effort  ;  but  to  ride  at  the  same  rate  against  such  a  wind  is  very 
difficult.  The  difference  is  due  entirely  to  the  resistance  of  the 
air.  In  the  first  case  this  resistance  is  zero,  since  rider  and  air 
are  relatively  at  rest ;  in  the  second  case  it  is  the  same  as  if  there 
were  no  wind  and  the  rider  were  going  at  the  rate  of  20  mi.  per 
hr.  TTiat  this  resistance  is  very  considerably  is  shown  by  the 
increase  of  effort  required  to  ride  against  the  wind. 

Falling  bodies  meet  with  a  rapidly  increasing  resistance  of  the 
air,  which  leaves  a  continually  diminishing  portion  of  their  weight 
unbalanced  to  cause  further  acceleration.  Compact  bodies  of 
considerable  density,  such  as  a  stone,  seldom  fall  far  enough  for  this 
resistance  to  become  appreciable ;  but  bodies  having  a  large  sur- 
face in  proportion  to  their  mass,  as  a  sheet  of  paper  or  a  leaf,  do 
not  fall  far  before  the  friction  equab  their  weight,  and,  as  there 
is  then  no  unbalanced  force,  there  is  no  further  acceleration.  It 
is  for  this  reason  that  the  velocity  of  raindrops  becomes  constant 
long  before  they  reach  the  ground. 

101.  Canae  of  the  Equal  Acceleration  of  Falling  Bodies.  —The 
equal  acceleration  of  falling  bodies  is  due  to  the  fact  that  weight 
is  proportional  to  mass.  For  example,  the  earth  attracts  equal 
portions  of  a  large  stone  and  a  small  one  equally  ;  and  the  total 
force  on  the  larger  (/>.  its  weight)  is  as  many  times  greater  than 
that  on  the  smaller  as  the  mass  of  the  one  is  greater  than  the  mass 
of  the  other.  Thus  the  two  forces  cause  equal  accelerations  be- 
cause they  are  proportional  to  the  masses  of  the  bodies  upon  which 
ihey  act. 

This  important  law  holds  for  all  forces.  Thus,  if  the  mass  of 
one  street  car  and  its  load  of  passengers  is  twice  that  of  another, 
twice  as  great  a  force  will  be  required  to  give  the  same  accelera- 
tion to  it  as  to  the  other.     It  should  be  noted  that  weight  is  not 


Falling  Bodies  79 

involved  in  this  illustration,  for  the  entire  weight  of  both  cars  is 
supported  by  the  track. 

102.  Motion  of  a  Sphere  on  an  Inclined  Plane.  —  When  a  ball  is 
on  an  inclined  plane  but  is  otherwise  unsupported,  its  motion  down 
the  plane  is  due  to  the  component  of  its  weight  acting  parallel  to 
the  plane  (Art.  66).  Since  this  component  is  constant  during  the 
descent  of  the  ball,  the  motion  is  uniformly  accelerated  (Art.  99)  ; 
but  the  acceleration  is  less  than  that  of  falling  bodies,  since  only 
part  of  the  weight  is  effective.  By  diminishing  the  inclination  of 
the  plane  the  acceleration  can  be  made  as  small  as  desired,  thus 
making  it  possible  to  determine  with  considerable  accuracy  the 
distances  traversed  by  the  ball  in  successive  seconds  {Exp.),  It 
will  be  found  by  trial  that  these  distances  are  in  the  ratio  of  the 
numbers  i,  3,  5,  7,  etc.  That  is,  if  x  denotes  the  distance  traversed 
in  the  first  second,  the  distances  traversed  during  the  second,  third, 
and  fourth  seconds  are  3^,  5  -r,  and  7  Xj  respectively.  The  whole 
distance  traversed  in  i  sec.  is  x,  in  2  sec.  4  ;c,  in  3  sec.  9  jc,  in  4 
sec.  16  AT,  etc.  It  is  evident  that  the  whole  distance  is  proportional 
to  the  square  of  the  time  ;  and  a  further  analysis  of  these  results 
leads  to  all  the  laws  of  uniformly  accelerated  motion. 

This  method  was  first  employed  by  Galileo  in  studying  the  laws 
of  falling  bodies. 

PROBLEMS 

1.  How  far  does  a  body  fall  during  the  first  second?  Account  for  the 
fact  that  this  distance  is  equal  to  half  the  acceleration. 

2.  (a)  What  is  the  velocity  of  a  falling  body  at  the  end  of  the  first  second  ? 
(/J)  How  far  does  it  fall  during  the  second  second?  (r)  Account  for  the 
difference  between  these  numbers. 

3.  What  is  the  velocity  of  a  falling  body  at  the  end  of  the  fifth  second  ? 

4.  How  far  does  a  body  fall  («)  in  5  sec?  {b)  in  6  sec?  (<:)  during  the 
sixth  second  ? 

5.  (a)  \Vhat  is  the  difference  between  the  distance  fallen  during  the  sixth 
second  and  the  velocity  at  the  beginning  of  that  second?  {b)  Is  this  differ- 
ence equal  to  that  found  in  the  second  problem?     Why? 

6.  A  stone  dropped  from  a  cliff  strikes  the  foot  of  it  in  3.5  sec  What  is 
the  height  of  the  cliff  ? 


8o  Dynamics 

7.  Why  b  it  that  the  increased  weight  of  a  body  when  taken  to  higher 
latitudes  causes  it  to  fait  faster,  while  at  the  same  place  a  heavy  body  falls 
no  faster  than  a  light  one? 

8.  When  a  train  is  leaving  a  station  its  acceleration  gradually  decreases 
to  zero,  although  the  engine  continues  to  pull  with  the  same  force  as  at  the 
start.     Explain. 

9.  Would  you  expect  the  motion  of  equally  smooth  and  perfect  spheres 
of  different  weight  and  material  to  be  equally  or  unequally  accelerated  on 
the  same  inclined  plane?    Give  reason  for  your  answer.    Try  the  experiment. 


m.   Projectiles 

103.  Projectiles. —  Any  body  moving  through  the  air  and  hav- 
ing a  component  velocity  not  imparted  by  its  weight  is  called  a 
projectile.  A  bullet  fired  from  a  gun,  an  arrow  shot  from  a  bow, 
and  a  ball  that  has  been  thrown  or  batted  are  examples  of  pro- 
jectiles. 

The  velocity  of  a  projectile  at  the  instant  when  the  force  that 
set  it  in  motion  ceases  to  act  is  called  its  initial  velocity.  Thus 
the  initial  velocity  of  a  bullet  is  its  velocity  at  the  muzzle  of  the 
gun.  The  initial  velocity  of  a  ball  when  thrown  is  its  velocity  at 
the  instant  it  leaves  the  hand  of  the  player. 

104.  Forces  acting  upon  Projectiles.  —  The  weight  of  a  projec- 
tile does  not  act  as  an  unbalanced  force  until  after  the  force  that 
imparts  the  initial  velocity  ceases.  Thus,  while  a  bullet  is  being 
driven  toward  the  muzzle  of  the  gun  by  the  pressure  of  the  expand- 
ing gases,  it  is  constrained  by  the  barrel  of  the  gun  to  go  in  a 
straight  path  ;  but  as  soon  as  it  leaves  the  muzzle,  it  is  freed  from 
this  restraint,  and  its  weight  acts  to  change  its  speed  and  direction 
of  motion.  The*  resistance  of  the  air  very  appreciably  affects  the 
motion  of  a  swiftly  moving  projectile,  as  a  rifle  ball ;  but  in  the 
cases  considered  in  this  boolc  it  is  disregarded. 

It  must  be  remembered  that  the  force  that  imparts  the  initial 
velocity  is  not  in  existence  during  the  flight  of  a  projectile.  It 
ceases  at  the  instant  the  projectile  is  launched.  Disregarding  the 
resistance  of  the  air,  the  weight  of  the  projectile  is  the  only  force 


Projectiles  8i 

acting  upon  it  during  its  flight,  and  is,  therefore,  the  sole  cause  of 
change  of  motion. 

105.  Effect  of  Unbalanced  Weight.  —  The  eff'ect  of  weight  upon 
the  motion  of  projectiles  is  illustrated  experimentally  by  releasing 
two  bodies  simultaneously  at  the  same  height, — one  with  a  con- 
siderable horizontal  velocity  imparted  by  a  sudden  push  or  blow, 
and  the  other  without  initial  velocity,  being  dropped  from  a  state 
of  rest  (Fig.  76).  The  two  bodies  will  always  be  found  to  reach 
the  floor  at  the  same  instant  {Exp).     The  experiment  illustrates 


A 


Fig.  76. 

the  fact  that  gravity  causes  the  same  acceleration  in  its  own  direc- 
tion whether  acting  upon  a  body  initially  at  rest  or  upon  a  body 
already  in  motion.  This  is  true  whatever  the  direction  of  the 
initial  velocity  may  be  ;  but  it  is  direcdy  evident  by  experiment 
only  when  the  initial  velocity  is  horizontal. 

A  projectile  has,  in  fact,  two  component  motions,  namely:  (i) 
the  initial  motion,  which  is  constant  in  magnitude  and  direction, 
since  there  is  no  force  acting  to  change  it  ;  and  (2)  the  uniformly 
accelerated  motion  due  to  gravity.  The  direction  of  the  first  is 
the  direction  of  projection  ;  the  direction  of  the  second  is  always 
vertically  down.  Neither  of  these  component  motions  interferes 
in  the  slightest  degree  with  the  other. 


82 


Dynamics 


"^ 

a' 

\ 

b' 

\ 

c» 

> 

\ 

Fig.  77. 


106.  Graphic  Representation  of  the  Path  of  a  Projectile.  —  The 
path  of  a  projectile  can  be  represented  graphically  by  compoiind- 

j  B  c  D  ^^^  its  two  motions  as  illustrated  in  Figs. 
77  and  78.  In  the  first  case  the  direc- 
tion of  projection  is  horizontal,  in  the 
second  it  is  obliquely  upward.  In  both 
figures  OA  denotes  the  initial  velocity 

g 

and  Oa  denotes  -   on  the  same  scale. 

2 

The  points  a\  b\  c\  etc.,  represent  the 
position  of  the  pro-  y) 

jectile   at   the   end  q^ 

of   successive    sec- 
onds.     A    smooth 
^'  curve  drawn  through 
these  points  repre- 
sents the  path  of  the  projectile. 

This  construction  indicates  that  if  the 
velocity  of  the  projectile  at  any  |K)int  of  its 
path  be  resolved  into  two  components,  — 
one  in  the  direction  of  projection  and  the 
other  vertical,  —  the  first  component  will 
be  the  initial  velocity,  and  the  second 
will  be  the  same  as  that  of  a  body  start- 
ing from  rest  and  falling  vertically  for  the 
same  time.  This  is  illustrated  in  the  fig- 
ures at  b\ 

107.  Projection  vertically  Upward.  —  While  rising  vertically, 
the  velocity  of  a  projectile  decreases  at  a  rate  equal  to  its  rate  of 
increase  in  falling,  provided  the  resistance  of  the  air  is  not  appre- 
ciable. Hence  the  times  of  rise  and  fall  are  equal  ;  and  both  the 
time  and  the  distance  can  be  computed  by  the  formulas  of  Art.  97. 

For  example,  if  a  stone  is  thrown  vertically  upward  with  an 
initial  velocity  of  49  m.  per  sec,  its  velocity  at  the  end  of  i  sec. 
is  49  —  9.8,  or  39.2  m.  per  sec. ;  and  at  the  end  of  2  sec.  it  is 


Fig.  78. 


The  Laws  of  Motion  83 

49  —  2  X  9.8,  or  29.4  m.  per  sec.  Its  time  of  rise  is  49  -^  9.8, 
or  5  sec. ;  for  at  the  end  of  that  time  its  velocity  would  be  zero. 
This  is  the  time  it  would  take  to  acquire  a  velocity  of  49  m.  per 
sec.  in  falling.  The  distance  that  the  stone  will  rise  is  found  by 
computing  the  distance  it  would  fall  in  the  same  time,  starting  from 
rest. 

PROBLEMS 

1.  (a)  What  is  the  force  that  causes  the  initial  velocity  of  an  arrow? 
(6)  How  long  does  it  act?  (c)  How  is  it  known  that  this  force  is  many  times 
greater  than  the  weight  of  the  arrow?  (</)  Is  the  acceleration  that  it  causes 
greater  or  less  than  that  due  to  gravity?  (<?)  How  long  does  this  accelera- 
tion continue  ?  (/)  What  would  be  the  motion  of  the  arrow  if  it  were  not 
acted  upon  by  any  force  during  its  flight? 

2.  Two  stones  are  thrown  to  the  same  height,  one  vertically,  the  other  ob- 
liquely.    Is  the  time  of  flight  the  same  for  both?     Explain. 

3.  A  stone  thrown  to  the  height  of  a  tree  reaches  the  ground  in  5  sec. 
from  the  time  of  starting.     How  high  is  the  tree? 

4.  A  body  is  thrown  horizontally,  with  an  initial  velocity  of  100  ft.  per  sec, 
from  the  top  of  a  tower  150  ft.  high.  At  what  distance  from  the  tower  will 
the  body  strike  the  ground? 

5.  An  arrow  is  shot  vertically  up  with  a  velocity  of  42  m.  per  sec.  (a)  How 
long  will  it  rise?     {d)  How  high  will  it  rise? 

6.  A  ball  is  thrown  upward  at  an  angle  of  30°  with  the  horizontal,  with 
an  initial  velocity  of  35  m.  per  sec.  (a)  What  is  the  time  of  its  flight  ? 
(3)  How  high  does  it  rise?  (<•)  How  far  from  the  starting  point  does  it 
strike  the  ground? 

Suggestion.  —  Resolve  the  initial  velocity  into  horizontal  and  vertical 
components.  The  first  component  is  constant ;  the  second  is  affected  by 
gravity,  just  as  it  would  be  if  the  first  component  did  not  exist. 


rv.   The  Laws  of  Motion 

108.  Balanced  and  Unbalanced  Forces.  —  The  effect  of  a  force 
is  always  to  cause  motion  or  to  change  the  existing  motion  of  the 
body  upon  which  it  acts,  unless  it  is  balanced  by  another  force  or 
other  forces  whose  tendency  is  to  produce  an  equal  and  opposite 
effect.  A  single  force  acting  upon  a  body  is  wholly  unbalanced. 
When  two  or  more  forces  act  simultaneously  upon  a  body,  the 


84  Dynamics 

unbalanced  force  is  their  resultant,  since  their  combined  effect  is 
the  same  as  their  resultant  would  produce  if  it  were  acting  alone 
{Exp.).  If  the  resultant  of  all  the  forces  acting  upon  a  body 
is  zero,  they  do  not  affect  its  state  of  rest  or  of  motion  in  any 
way ;  i.e.  the  body  behaves  as  it  would  if  no  force  at  all  were 
acting  upon  it. 

ITie  general  laws  of  dynamics,  or,  as  they  are  generally  called, 
the  laws  of  motion^  are  concise  and  definite  statements  of  the 
behavior  of  bodies  when  acted  upon  by  unbalanced  forces  and 
when  not  acted  upon  by  such  forces.  Throughout  the  following 
discussion  of  these  laws  the  word  force^  when  unqualified,  must 
always  be  understood  to  mean  unbalanced  or  resultant  force. 

109.  The  Effect  of  a  Constant  Force.  —  Any  constant  {unbal- 
anced) force  acting  upon  any  body  causes  uniform  acceleration  of 
its  motion  in  the  direction  in  which  the  force  acts.  The  weight  of 
a  body,  when  acting  alone,  is  such  a  force  ;  and  it  causes  a  con- 
stant acceleration  of  9.8  m.  per  sec.  per  sec.  in  its  own  direction 
whether  it  acts  upon  a  body  initially  at  rest  or  upon  a  body  hav- 
ing any  initial  velocity  in  any  direction  (Arts.  105-107). 

The  acceleration  of  a  given  mass  is  proportional  to  the  {unbal- 
anced) force  acting  upon  it.  This  law  is  illustrated  by  the  motion 
of  a  sphere  on  a  plane  when  inclined  at  different  angles.  The 
effective  {i.e.  unbalanced)  component  of  the  weight  of  the  ball 
varies  with  the  inclination  of  the  plane  (Arts.  66  and  102)  ;  and 
it  will  be  found  by  trial  that  the  acceleration  is  proportional  to 
this  component  (Exp.).  Thus  by  doubling  the  height  of  the 
plane  {BC,  Fig.  41),  the  unbalanced  component  of  the  weight  is 
doubled,  and  the  ball  will  roll  twice  as  far  in  the  same  time  as 
before,  showing  that  the  acceleration  has  been  doubled  (Law  V, 
Art.  93). 

110.  Effect  of  a  Variable  Force.  — When  the  force  acting  upon 
a  body  is  not  constant  but  varies  from  moment  to  moment,  the 
acceleration  at  any  instant  is  proportional  to  the  {unbalanced) 
force  at  that  instant. 

Illustrations  of  this  law  in  daily  life  are  numerous.     (See  the 


The  Laws  of  Motion  85 


discussion  of  the  motion  of  a  falling  leaf,  second  paragraph  of 
Art.  100,  and  of  the  motion  of  a  street  car,  second  paragraph  of 
Art.  91.)  We  can  now  understand  why  the  speed  of  a  street  car 
does  not  increase  as  long  as  the  motor  is  running.  The  friction 
(resistance  of  the  air,  etc.)  rapidly  increases  as  the  speed  increases, 
leaving  a  constantly  diminishing  unbalanced  force  to  accelerate 
the  motion  of  the  car.  When  the  total  resistance  of  friction  is 
equal  to  the  driving  force  due  to  the  motor,  the  resultant  force  upon 
the  car  is  zero,  and  its  speed  is  then  constant  (acceleration  zero). 

111.  Relation  between  Force  and  Acceleration  ;  Mass  Constant. 
—  The  three  laws  stated  in  the  two  preceding  articles  are  included 
in  the  following  :  — 

The  acceleration  of  a  body  is  always  proportional  to  the  unbal- 
anced force  acting  upon  it,  and  takes  place  in  the  direction  in  which 
the  force  acts. 

The  law  holds  whatever  the  relative  direction  of  the  force  and 
the  motion  of  the  body  may  be.  Three  cases  arise  as  follows : 
(i)  If  the  force  acts  in  the  direction  of  motion,  the  velocity  will 
be  increasing  but  constant  in  direction;  (2)  if  the  force  and  the 
motion  are  in  opposite  directions,  the  velocity  will  be  decreasing 
but  constant  in  direction ;  (3)  if  the  force  acts  at  an  angle  to  the 
direction  of  motion,  the  velocity  will,  in  general,  be  changing  both 
in  magnitude  and  direction.  (What  illustrations  of  the  three  cases 
are  afforded  by  falling  bodies  and  projectiles?) 

The  law  expresses  the  interesting  fact  that  an  unbalanced  force, 
however  small,  acting  upon  any  mass,  however  great,  will  move  it 
or  will  change  its  existing  motion.  The  change  of  motion  may, 
indeed,  be  very  slow,  but  it  will  be  none  the  less  certain.  On  the 
other  hand,  to  impart  a  very  great  velocity  to  a  body  in  a  very 
short  time  requires  great  force  even  though  the  mass  of  the  body 
be  small.  For  example,  the  pressure  of  the  expanding  gases 
behind  a  ten-pound  cannon  ball  in  the  act  of  firing  it  is  several 
hundred  thousand  pounds.  Since  the  acceleration  is  proportion- 
ately great,  the  ball  leaves  the  muzzle  of  the  cannon  with  a  very 
high  velocity. 


86  Dynamics 

If  two  forces,  acting  upon  the  same  mass  or  upon  equal  masses, 
are  denoted  by  /,  and  y^  and  the  accelerations  that  they  produce 
by  ai  and  a^  respectively,  then  the  law  states  that 

/, :  yi  :  :  a, :  <j^     (mass  constant)  (i  i ) 

Applying  /^  and  <?,  to  the  case  of  a  freely  falling  body,  /^  is 
the  weight  of  the  body,  «/,  and  a^  is  g.  Hence  in  this  case  the 
proportion  becomes 

/:w::a:g.  (12) 

From  this  proportion  any  one  of  the  quantities/,  7a,  and  a  can 

be  computed  if  the  other  two  are  known,  g  being  taken  as  980  cm. 

or  32.15  ft.  per  sec.  per  sec.    Thus,  if  a  sphere  weighing  800  g. 

is  placed  upon  a  plane  inclined  at  an  angle  such  that  the  effective 

component  of  its  weight  (allowing   for   the   force   necessary  to 

produce  rotation)  is  200  g.,  its  acceleration  will  be  given  by  the 

proiK>rtion  o  o 

"^    *  200  :  800  ::  a  :  980. 

112.  Relation  between  Force  and  Mass ;  Acceleration  Constant. 
—  Tke  {unbalanced)  force  necessary  to  produce  a  given  acceleration 
is  proportional  to  the  mass  of  thf  body  upon  which  the  force  acts. 
Expressed  algebraically  the  law  is 

fx'.ft'.'.  /«i :  Wj.     (acceleration  constant)  (13) 

The  equal  acceleration  of  falling  bodies  (Art.  101)  serves  as  the 
simplest  and  best  illustration  of  this  law,  since  weight  is  always 
proportional  to  mass.  But  experiments  in  which  the  acceleration 
is  not  due  to  gravity  are  instructive,  even  if  they  do  not  prove  the 
definite  relation  expressed  in  the  law.  The  following  experiment 
is  of  this  sort. 

Suspend  a  lead  or  an  iron  ball  an  inch  or  more  in  diameter  by 
a  cord  one  or  two  meters  long,  and  suspend  a  cork  of  about  the 
same  size  by  a  cord  of  equal  length.  With  a  swift  horizontal  swing 
of  the  arm,  strike  the  cork  with  the  open  palm.  Strike  the  lead 
ball  in  the  same  way,  with  equally  rapid  motion  of  the  hand.  (A 
small  board  may  be  placed  a  foot  or  more  beyond  the  ball  to  stop 


The  Laws  of  Motion  87 

it.)  The  force  exerted  upon  the  ball  is  very  considerable,  while 
that  upon  the  cork  is  almost  inappreciable,  although  it  is  started 
with  equally  accelerated  motion.  It  is  evident  that  the  difference 
is  not  due  to  the  greater  weight  of  the  lead  ball,  for,  in  the  vertical 
position,  its  weight  is  entirely  supported  by  the  cord.  Neither  is 
the  difference  due  to  friction  or  any  other  force.  The  lead  ball 
requires  the  greater  force  to  start  it  because  its  mass  is  greater. 
Having  greater  mass  it  has  greater  inertia,  and  this  requires  pro- 
portionally greater  force  for  an  equal  acceleration. 

113.  The  Comparison  of  Masses  by  the  Inertia  Test.  —  It 
follows  from  the  law  stated  in  the  preceding  article  that,  if  equal 
forces  act  upon  unequal  masses,  the  acceleration  of  the  smaller 
mass  will  be  the  greater.  The  acceleration  will,  in  fact,  be  in- 
versely proportional  to  the  masses.  Hence,  if  equal  forces  impart 
equal  accelerations  to  two  masses,  the  masses  are  equal  {Exp.). 

This  is  the  inertia  test  or  acceleration  test  of  the  equality  of 
two  masses.  Masses  that  are  equal  by  the  inertia  test  are,  of 
course,  equal  by  the  usual  weight  test  also.  The  inertia  test  is 
more  fundamental  and  scientifically  more  significant,  —  a  fact  that 
can  hardly  be  appreciated  by  students  of  elementary  physics ;  but 
the  weight  test  is  more  accurate  and  much  more  convenient,  and 
hence  is  used  exclusively  in  scientific  work  as  well  as  in  daily  life. 

Laboratory  Exercise  20. 

114.  The  Element  of  Time  in  the  Effect  of  Force.  —  Since 
change  of  velocity  is  proportional  to  the  time  during  which  a  given 
acceleration  continues  {v  =  at),  it  follows  that  the-  change  of 
velocity  produced  by  a  constant  force  is  proportional  to  the  time 
during  which  the  force  acts.  Some  forces  are  very  great,  but  act 
for  an  extremely  short  time,  as  the  blow  of  a  hammer,  or  the 
force  exerted  by  a  bullet  in  penetrating  a  board.  The  time  is  so 
extremely  brief  in  the  latter  case  that  a  bullet  fired  through  a  door 
standing  ajar  will  scarcely  disturb  it,  although  it  can  be  swung  with 
a  light  push  of  the  finger. 

These  ideas  are  illustrated  by  a  simple  experiment  with  a  small 
coin  and  a  calling  card.     The  friction  between  them  when  the  coin 


88  Dynamics 

is  placed  on  the  card  is  sufficient  to  impart  the  motion  of  the  card 

to  the  coin  when  the  card  is  moved  slowly  about ;  but,  when  it  is 

very  suddenly  started,  the  coin  is   left   behind.     This   is   neatly 

shown  by  placing  the  card  and  coin  on 

a  finger  (Fig.  79)  and  suddenly  snapping 

the  card  in  a  horizontal  direction.     If  the 

blow  is  successfully  aimed,  the  card  will 

fly  from  under  the  coin,  leaving  it  at  rest 

on  the  finger,  friction  being  insufficient 

to  impart  appreciable  motion  to  the  coin 

in  so  short  a  time.    (Try  the  experiment.) 
Fig.  79. 

PROBLEMS 

1.  The  acceleration  of  any  falling  body  is  proportional  to  its  weight  in 
different  latitudes  and  at  different  altitudes  (Art.  98)  ;  but  all  bodies  at  the 
same  place  fall  with  equal  acceleration,  whatever  their  weight  (unless  re- 
tarded by  the  air).     Explain. 

2.  A  bullet  fired  through  a  plate  glass  window  will  often  make  a  smooth 
hole  without  cracking  the  glass.     Explain. 

3.  A  nail  can  be  driven  by  striking  it  with  a  hammer,  but  not  by  pressing 
the  hammer  steadily  against  it.     Explain. 

4.  Gravity  upon  the  moon  is  one  sixth  as  great  as  upon  the  earth.  G)m- 
pute  the  acceleration  of  a  falling  body  upon  the  moon. 

5.  Gravity  upon  the  sun  is  27.6  times  as  great  as  upon  the  earth.  Com- 
pute the  acceleration  of  a  falling  body  upon  the  sun. 

6.  How  far  would  a  body  fall  during  the  first  second  (a)  upon  the  moon  ? 
(^)upon  the  sun? 

7.  (a)  Would  the  mass  of  a  given  body  be  the  same  upon  the  sun  or 
the  moon  as  upon  the  earth?     (6)  Would  its  inertia  be  the  same? 

8.  Would  it  take  less  powder  to  fire  a  cannon  ball  with  a  given  velocity 
upon  the  moon  than  it  would  upon  the  earth  ? 

9.  Is  it  harder  for  horses  to  start  a  loaded  wagon  or  to  keep  it  in  uniform 
motion?     Give  reasons. 

10.  Why  does  a  ball  player  move  his  hands  quickly  backward  in  the  act 
of  catching  a  swift  ball  ? 

11.  An  unbalanced  force  of  25  g.  acts  on  a  mass  of  80  g.  What  is  the 
acceleration  ? 

12.  WTiat  force  is  required  to  impart  an  acceleration  of  15  cm.  per  sec. 
per  sec.  to  a  mass  of  100  g.? 


The  Laws  of  Motion  89 

115.  The  Law  of  Inertia.  —  Since  change  of  motion  results  only 
from  the  action  of  an  applied  force,  in  accordance  with  the  pre- 
ceding laws,  matter  is  said  to  be  passive  or  inert,  and  the  property 
thus  manifested  is  called  inertia^  (Art.  9).  The  law  of  inertia  is 
as  follows  :  — 

Every  body  continues  in  its  state  of  rest  or  of  uniform  motion  in  a 
straight  line  unless  compelled  to  change  that  state  by  an  external 
force. 

This  law  follows  as  a  corollary  from  the  law  stated  in  Art.  1 1 1 ; 
for  the  acceleration  must  be  zero  when  the  unbalanced  force  is 
zero,  since  the  two  are  proportional,  and,  with  zero  acceleration, 
motion  remains  constant  both  in  magnitude  and  direction.  It  is 
impossible  to  prove  the  law  of  inertia  by  direct  experiment,  since 
no  body  can  be  freed  from  the  action  of  all  forces  ;  but  the  indirect 
evidence  of  its  truth  is  conclusive  (see  Art.  9).  Astronomical 
observations  on  the  motions  of  the  moon  and  the  planets  confirm 
all  the  laws  of  motion  as  well  as  the  law  of  gravitation. 

116.  The  Law  of  Mutual  Action.  —  To  every  action  there  is  an 
equal  and  opposite  reaction  ;  or,  the  mutual  actions  of  two  bodies 
are  ahvays  equal  and  in  opposite  directions. 

Illustrations  of  this  law  were  considered  in  Art.  12,  and  there 
have  been  numerous  applications  of  it  in  the  study  of  fluids  and 
the  statics  of  solids.  The  law  holds  for  all  forces,  whether  bal- 
anced or  unbalanced.  Force  exists  only  through  the  mutual 
action  of  two  bodies  (or  two  parts  of  the  same  body,  which  amounts 
to  the  same  thing).  Thus  every  force  is  one  of  a  pair  of  equal  and 
opposite  forces,  exerted  by  each  of  two  bodies  on  the  other.  The 
two  forces  do  not  balance  each  other,  since  they  act  upon  different 
bodies  ;  but  either  or  both  may  be  balanced  by  other  forces.  For 
example,  the  pressure  exerted  by  a  bat  upon  a  ball  in  striking  it  is 
unbalanced  and  imparts  motion  to  the  ball.  The  reaction  of  the 
ball  upon  the  bat  is  also  unbalanced  and  checks  the  motion  of  the 

1  The  pupil  should  avoid  the  misconception  that  the  inertia  of  matter  is  an 
active  agent  opposing  and,  in  some  sense,  neutralizing  the  effect  of  force.  Inertia 
is  not  force,  nor  does  it  ever  balance  a  force. 


9©  Dynamics 

bat.  ^Vhen  a  piece  of  iron  placed  on  an  anvil  is  struck  with  a 
hammer,  the  blow  of  the  hammer  is  balanced  by  the  equal  and 
opposite  pressure  of  the  anvil,  both  acting  on  the  piece  of  iron  ; 
hence  the  iron  remains  at  rest.  When  a  person  jumps  from  a 
boat,  the  reaction  on  the  boat  is  unbalanced  and  pushes  the  boat 
in  the  opposite  direction  from  that  in  which  the  person  jumps ; 
but,  in  jumping  from  a  rock,  the  reaction  upon  the  rock  is  balanced 
by  the  friction  between  it  and  the  ground,  and  it  remains  at 
rest. 

It  is  often  supposed  that  the  motions  of  animals  and  self-pro- 
pelling machines  are  independent  of  applied  force,  since  they 
*'  make  themselves  go."  The  motion  of  all  matter,  animate  or 
inanimate,  is  in  accordance  with  the  same  laws  of  motion.  The 
real  difference  between  the  conditions  of  motion  of  a  stone  and 
an  animal  or  an  engine,  is  that  the  latter  can  cause  the  applied 
forces  that  move  them,  and  a  stone  cannot.  For  example,  when 
a  boy  jumps,  he  pushes  vigorously  with  his  feet  downward  and 
backward,  against  the  ground  with  a  force  much  greater  than  his 
weight.  The  ground  reacts  with  an  equal  and  opposite  force  upon 
the  boy,  and  it  is  this  reaction  that  enables  him  to  spring  upward 
and  forward.  A  similar  reaction  of  the  ground  takes  place  with 
every  step  in  running.  The  runner  also  leans  forward,  in  which 
position  his  weight  pulls  his  body  forward,  as  in  the  act  of  falling. 
When  one  attempts  to  start,  stop,  or  turn  quickly  while  walking  on 
ice,  friction  is  too  slight  to  cause  the  necessary  reaction  upon  the 
feet,  and  this  results  in  a  fall.  A  bird  in  flying  pushes  the  air 
doAWiward  and  backward  with  its  wings ;  the  reaction  of  the  air 
upward  and  forward  sustains  the  bird  in  its  flight.  The  mutual 
action  between  the  driving  wheels  of  an  engine  and  the  rails  is 
different  from  that  between  the  car  wheels  and  the  rails,  as  is 
shown  by  the  fact  that  the  former  sometimes  slip,  spinning  round 
and  round,  while  the  latter  never  do.  A  driving  wheel  exerts  a 
strong  backward  push  on  the  rail,  and  slipping  is  prevented  only 
by  friction ;  the  forward  reaction  of  the  rail  on  the  wheel  is  essen- 
tial to  the  motion  of  the  engine,  and  is  an  external  force. 


The  Laws  of  Motion  91 


117.  Relative  Velocities  due  to  Unbalanced  Action  and  Re- 
action ;  Momentum.  —  When  the  mutual  actions  of  two  bodies  are 
unbalanced,  their  accelerations  are  inversely  proportional  to  their 
masses,  since  the  forces  exerted  by  each  upon  the  other  are  equal 
(Art.  113,  first  paragraph)  ;  and,  if  the  bodies  are  initially  at  rest, 
their  velocities  will  also  be  inversely  proportional  to  their  masses. 
Thus  when  a  man  jumps  from  a  boat  that  weighs  three  times  as 
much  as  himself,  the  boat  is  pushed  back  with  a  velocity  one  third 
as  great  as  the  forward  velocity  of  the  man.  A  rifle  "  kicks  "  when 
fired,  because  the  gas  from  the  burned  powder  presses  back  on  the 
rifle,  as  well  as  fonvard  on  the  bullet,  and  with  equal  force.  The 
velocities  of  the  rifle  and  the  bullet  are  inversely  proportional  to 
their  masses. 

Let  nty  and  W2  denote  the  masses  of  two  bodies  initially  at  rest, 
and  Vx  and  v.^  their  respective  velocities  imparted  by  mutual  action, 
the  bodies  being  free  to  move ;  then  nti  -.m^wv^:  z/i,  or  tn-^Vx 
«=  W2?v  The  product  of  the  mass  of  a  body  and  its  velocity  is 
called  its  momentum.  Hence  bodies  initially  at  rest  and  free  to 
move  acquire  equal  momenta  in  opposite  directions  as  a  result 
of  their  mutual  actions.  Thus  the  momentum  of  a  rifle  in  its  recoil 
is  equal  to  the  momentum  of  the  bullet  as  it  leaves  the  muzzle. 
When  a  moving  body  strikes  a  body  at  rest,  and  their  mutual 
actions  are  unbalanced,  the  one  loses  as  much  momentum  as  the 
other  gains. 

118.  Newton's  Laws  of  Motion.  —  The  fundamental  laws  of 
motion  considered  in  the  preceding  articles  are  restated  here  for 
convenient  reference  :  — 

I.  Every  body  continues  in  its  state  of  rest  or  of  uniform  motion 
in  a  straight  line^  except  in  so  far  as  it  is  compelled  by  external 
forces  to  change  that  state. 

II  a.  The  acceleration  of  a  body  is  proportional  to  the  unbalanced 
force  acting  upon  it^  and  is  in  the  direction  of  that  force  ;  or, 

f'.fi'.'.ai'.a^.     (mass  constant) 

II  b.    The  unbalanced  force  necessary  to  produce  a  given  accelera- 


92  Dynamics 

tion  is  proportional  to  the  mass  of  the  body  upon  which  the  force 

€Uts  ;  or, 

fx'fi''  '"i  •  '"s«     (acceleration  constant) 

III.  To  nrr}'  action  there  is  an  equa/  ami  opposite  reaction  ;  or 
the  mutual  actions  of  two  bodies  are  always  equal  and  in  opposite 
directions. 

The  laws  numbered  I  and  III  are  known  as  Newton's  first  and 
third  laws  of  motion  respectively  ;  Ila  and  lib  are  together  equiv- 
alent to  his  second  law  of  motion.  These  laws  constitute  a  com- 
plete statement  of  the  relation  between  matter  and  force.  The 
first  and  second  were  discovered  by  Galileo  in  studying  the  motion 
of  falling  l>odies  and  projectiles.  The  third  law  was  also  known  to 
others  before  Newton.  They  are  called  Newton's  laws  because 
he  was  the  first  to  state  them  in  their  present  form. 

Note  to  the  Teacher.  —  The  second  law  of  motion  as  stated  by  Newton 
can  be  derived  Irom  II  a  and  lib  as  follows:  According  to  II  a,  /oc  a  (/«  =  con- 
stant);  according  to  II  b,  /ac  m  (a  =  constant).  Hence /oc  ma,  and  _/?oc  mat. 
When  the  initial  velocity  is  zero,  the  latter  may  be  written  //«  mv.  The  equality 
of  ft  and  mv  requires  the  introduction  of  the  dyne  as  the  unit  of  force.    With  the 

gravitational    unit    of    force,  the    equations   are  /=  —  and  ft  = These 

equations  express  the  second  law  in  mathematical  form.  Stated  in  words,  it  is : 
Change  of  momentum  is  proportional  to  the  appHed  force  and  to  the  time  during 
which  it  acts,  and  takes  place  in  the  direction  of  the  force. 


PROBLEMS 

1.  Two  boys,  A  and  B,  are  pulling  upon  the  ends  of  a  rope.  A  pulls  B 
along.     Is  he  pulling  harder  (/.^.,  with  greater  force)  than  B?     Explain. 

2.  Two  boats  are  afloat  some  distance  apart,  and  at  rest.  A  man  sitting 
in  one  of  them  hauls  in  a  rope  attached  to  the  other.  Describe  and  explain 
the  motions  of  the  boats,  assuming  them  to  be  {a)  of  equal  mass  (including 
the  mass  of  whatever  is  in  the  boats;  ;   (Z^)  of  uneciual  mass. 

3.  Why  does  stamping  remove  mud  from  the  shoes  ? 

4.  Why  does  beating  a  carpet  remove  dust  from  it  ? 

5.  Why  can  the  handle  be  tightened  in  the  head  of  an  ax  (a)  by  striking 
the  end  of  the  handle  against  a  log?  (/')  by  holdingthe  ax  at  rest  and  strik- 
ing the  end  of  the  handle  with  a  hammer  ? 


Curvilinear   Motion  93 

6.  Two  battle  ships  are  in  an  engagement,  with  one  in  pursuit  of  the 
other.  Does  the  reaction  of  the  guns  in  liring  aid  or  hinder  the  speed  of  the 
pursuer?   of  the  pursued? 

7.  How  should  a  person  handle  his  body  to  avoid  a  fall  when  alighting 
from  a  rapidly  moving  car  ? ,   Explain. 

V.  Curvilinear  Motion 

119.  Cause  of  Curvilinear  Motion.  —  We  have  learned  that  the 
weight  of  a  projectile  causes  its  path  to  curve  downward,  unless 
the  motion  is  vertical.  Any  unbalanced  force  acting  upon  a  body 
at  an  angle  to  its  direction  of  motion  produces  a  similar  effect ; 
i.e.  causes  the  path  of  the  body  to  curve  toward  the  direction  in 
which  the  force  acts.  A  stone  tied  to  a  string  and  whirled  in  a 
circle  round  the  hand  is  a  familiar  illustration.  The  motion  of  the 
stone  in  a  circle  is  (Uie  to  the  continued  inward  pull  of  the  string 
upon  it.  If  the  stone  is  released  at  any  point  of  its  path,  it  con- 
tinues in  the  direction  of  its  motion  at  the  instant  of  release, 
except  in  so  far  as  its  motion  is  then  affected  by  its  weight. 

The  following  experiment  affords  a  better  illustration  :  A  wooden 
ball  tied  to  a  string  is  rolled  round  in  a  circle  on  the  top  of  a  large 
table  or  on  the  floor.  When  released  at  any 
point  of  its  path,  it  continues  in  the  direction 
in  which  it  was  moving  at  that  instant.  Thus, 
if  released  at  A  (Fig.  80),  its  path  will  be  AB, 
a  line  tangent  to  the  circle  at  A  {Exp.).  The 
law  illustrated  by  the  experiment  is  general. 
A  body  moves  in  a  curved  path  only  when  acted  ^^'   °* 

upon  by  an  unbalanced  force  directed  toward  the  inside  of  the  curve. 

A  force  acting  upon  a  body  so  as  to  change  its  direction  of  mo- 
tion is  called  a  centripetal  force,  because  it  acts  toward  the  center 
of  the  curved  path  (from  the  Latin  centrum,  center,  ^n^petere,  to 
seek).  A  centripetal  force  may  act  at  right  angles  to  the  direc- 
tion of  motion,  or  obliquely  forward  or  backward.  The  three 
cases  are  illustrated  at  M,  N,  and  L  respectively  in  Fig.  81,  which 
represents  the  path  of  a  projectile.     Strictly  speaking,  the  centripe- 


94 


Dynamics 


/ 


Fig.  8i. 


tal  force  at  L  and  N  is  the  component  of  weight  acting  at  right 

angles  to  the  path,  />.  the  component/.  The  tangential  compo- 
nent, Ty  acts  opposite  to 
#  ^^       ^v,^^  the   direction   of  motion 

at  Z,  causing  decrease  of 
speed,  and  in  the  direc- 
tion of  motion  at  W,  caus- 
ing increase  of  speed. 
The  centripetal  compo- 
nent at  L  and  N  and  the 
entire  weight  at  M  cause 
change  of  direction  only. 
Uniform  motion  in  a  circle  is  due  to  a  constant  centripetal  force, 

which  always  acts  toward  the  center  of  the  circle  and  at  right  angles 

to  the  direction  of  motion. 

120.  Laws  of  Centripetal  Force.  —  It  can  be  shown  either  by 
experiment  or  by  mathematical  analysis  based  on  the  second  law 
of  motion  that  centripetal  force  is  (i)  proportional  to  the.  mass 
of  the  body,  (2)  proportional  to  the  square  of  the  velocity,  and 
(3)  inversely  proportional  to  the  radius  of  curvature  of  the  path. 

The  effect  of  the  mass  of  the  body  can  be  shown  by  whirling 
unequal  masses  with  equal  rapidity,  using  strings  of  equal  length ; 
and  the  effect  of  velocity,  by  whirl- 
ing the  same  body  more  and  less 
rapidly.     (Try  it.) 

121.  Illustrations  of  Centripetal 
Force.  —  If  a  ball  or  other  object 
is  suspended  by  a  string  from  a  fixed 
support  and  started  in  a  horizontal 
circle  (Fig.  82),  it  will  continue  to 
revolve  in  a  slowly  diminishing  circle 
(more  accurately  a  spiral)  for  sev- 
eral minutes  {Exp.).  The  decrease 
in  the  size  of  the  circle  is  due  to 
friction,  chiefly  of  the  air,  and  may  Fig.  83. 


Curvilinear  Motion  95 

be  disregarded.  If  all  friction  could  be  removed,  the  motion 
would  continue  indefinitely  without  change.  Disregarding  fric- 
tion, the  ball  is  acted  upon  by  two  forces ;  namely,  its  weight, 
IV,  and  the  tension,  T,  of  the  cord.  The  vertical  component 
of  the  tension,  v,  is  equal  to  IV  and  balances  it ;  the  horizontal 
component,  /,  is  unbalanced,  and  is  directed  toward  the  center 
of  the  circle.  The  component  /  is  the  centripetal  force  that 
causes  the  circular  motion  of  the  ball.  Since  this  force  acts  at 
right  angles  to  the  direction  of  motion,  it  has  no  effect  on  the 
speed  (Art.  119,  last  paragraph). 

In  rounding  a  curve  a  bicycle  rider  brings  the  necessary  cen- 
tripetal force  to  bear  upon  his  body  by  leaning  toward  the  inside 
of  the  curve.  Let  C  (Fig.  S^)  denote  the  cen- 
ter of  the  curved  path  of  the  rider  and  wheel, 
and  OB  the  inclination  of  the  wheel.  The  wheel 
exerts  an  oblique  pressure  upon  the  ground  in 
the  direction  OB  (as  is  shown  by  the  fact  that 
when  a  wheel  slips  in  turning  a  curve  it  always 
slips  outward).  The  reaction  of  this  force  is  an 
equal  pressure  of  the  ground  against  the  wheel 
in  the  direction  BO,  and  is  denoted  in  the  figure  fig.  83. 

by  OPy  as  if  it  were  applied  at  the  center  of  gravity.  This 
oblique  inward  pressure  of  the  ground  may  be  considerably 
greater  than  the  weight  of  the  wheel  and  rider,  which  is  denoted 
by  OIV.  The  vertical  component  of  OP  is  equal  to  the  weight 
of  the  rider  and  wheel  and  balances  it.  The  horizontal  compo- 
nent,/, is  the  centripetal  force  upon  the  rider  and  wheel. 

122.  Inertia  shown  in  Curvilinear  Motion.  —  The  tendency  of 
moving  bodies  to  move  in  a  straight  line,  as  stated  in  the  law  of 
inertia,  is  shown  by  the  fact  that  curvilinear  motion  continues 
only  so  long  as  a  centripetal  force  acts  to  maintain  it.  From  the 
instant  that  centripetal  force  ceases  to  act  upon  a  body,  it  con- 
tinues in  a  straight  line,  or,  to  use  a  familiar  expression,  it  "  flies 
off  at  a  tangent."  Thus  motion  in  a  curve  is  explained  by  describ- 
ing the  centripetal   force  that  causes  it ;   while  "  flying  off  at  a 


96  Dynamics 

tangent "  is  accounted  for  by  noting  the  absence  of  centripetal 
force,  this  behavior  being  merely  a  result  of  the  inertia  of  matter. 

To  illustrate  :  When  a  carriage  is  driven  round  a  corner,  its 
tend^cy  is  to  continue  in  a  straight  line ;  hence  the  wheels  tend 
to  slip  over  the  ground  toward  the  outside  of  the  curve.  Ordi- 
narily friction  is  sufficient  to  prevent  the  shpping,  and  this  causes 
the  ground  to  react  with  an  inward  pressure  on  the  wheels.  But, 
since  the  wheels  cannot  slide,  the  tendency  of  the  carriage  to 
continue  in  a  straight  line  results  in  a  tendency  to  overturn  out- 
ward. This  tendency  is  opposed  by  the  weight  of  the  carriage, 
which,  under  ordinary  circumstances,  acts  as  a  sufficient  cen- 
tripetal force  to  bring  the  carriage  safely  round  the  curve.  If, 
however,  the  motion  is  very  rapid  and  the  curve  sharp,  this  cen- 
tripetal force  may  be  insufficient ;  in  which  case  the  carriage  will 
overturn,  not  because  of  a  force  acting  to  oi'erturn  it,  but  because 
thf  centripetal  component  of  weight  is  insufficient  to  p?vduce  the 
necessary  change  of  direction.  The  behavior  of  water  on  a  rotating 
grindstone  is  a  further  illustration.  The  water  is  held  to  the  stone 
by  adhesion ;  but  when  the  speed  reaches  a  certain  value,  the 
adhesion  is  no  longer  sufficient  to  carry  the  water  round  in  the 
curved  path,  and  it  flies  off.  Mud  flies  from  the  wheels  of  a 
rapidly  moving  carriage  for  the  same  reason. 

123.  Centrifugal  Force.  —  When  a  stone  is  whirled  at  the  end 
of  a  string,  it  exerts  an  outward  pull  through  the  string  upon  the 
hand,  which  is  the  equal  and  opposite  reaction  of  the  pull  that  the 
hand  exerts  upon  the  stone.  The  reaction  is  called  centrifugal 
force,  since  its  direction  is  outward  from  the  center  (from  the 
Latin  centrum,  ^xiA  fugere,  to  flee).  The  centrifugal  force  is 
exerted  upon  the  hand,  and  tends  to  pull  it  (not  the  stone) 
outward. 

It  is  a  common  but  wholly  mistaken  idea  that  centrifugal  force 
causes  or  tends  to  cause  bodies  to  leave  a  curved  path  and  fly  off 
or  overturn.  When  a  carriage  rounds  a  comer,  the  only  cen- 
trifugal force  in  action  is  the  outward  pressure  of  the  wheels  upon 
the  ground.     Centrifugal  force  does  not  act  upon  the  body  mov- 


Curvilinear  Motion 


97 


ing  in  the  curved  path;  hence  it  cannot  under  any  circumstances 
affect  the  motion  of  that  body.  The  confusion  of  thought  with 
reference  to  centrifugal  force  arises  from  the  fact  that  the  term 
was  originally  appHed  to  a  fictitious  force,  and  is  still  used  in  this 
sense  in  unscientific  language.  This  fictitious  force  is  supposed 
to  act  on  the  moving  body  and  to  be  the  cause  of  its  tendency  to 
"  fly  off  at  a  tangent."  No  such  cause  exists,  and  the  assumption 
that  it  does  exist  only  leads  to  a  misunderstanding  of  the  whole 
subject.  Centrifugal  force  in  the  only  sense  in  which  the  word 
should  be  used,  need  not  be  mentioned  in  discussing  curvilinear 
motion,  for,  although  it  is  a  real  force,  it  does  not  act  upon  the 
body  whose  motion  is  under  consideration. 

PROBLEMS 

1.  Why  is  the  curvature  of  the  path  of  a  projectile  the  greatest  at  its 
highest  point  ?     (See  Fig.  8i.) 

2.  Explain  what  would  happen  if  a  bicycle  rider  failed  to  lean  inward 
sufficiently. 

3.  Why  are  curves  in  bicycle  race  tracks  steeply  inclined  toward  the 
center  ? 

4.  If  a  boy,  while  running,  wishes  to  change  his  direction  suddenly,  as  in 
dodging,  how  does  he  handle  his  body  ?     Explain. 

5.  A  bucket  of  water  can  be  whirled  in  a  vertical  circle,  the  bucket  being 
inverted  at  the  top  of  the  circle,  without  any  of  the  water  spilling.  (Try  it.) 
Explain. 

6.  Draw  three  figures  similar  to  Fig.  82  ;  one  representing  the  case  where 
the  deflection  of  the  cord  from  the  vertical  is  only  a  few  degrees,  one  where 
it  is  40*^  to  50^  and  one  where  it  is  70°  to  80°.  The  angles  need  not  be 
measured,  but  the  forces  are  to  be  represented  to  the  same  scale  in  all.  The 
tension  must  always  be  taken  of  such  magnitude  that  its  vertical  component 
is  equal  to  the  weight  of  the  ball.  How  does  the  centripetal  force  vary  as 
the  cord  becomes  more  nearly  horizontal?  Would  it  be  possible  to  swing 
the  ball  fast  enough  to  bring  the  cord  to  a  horizontal  position?  Give  reason 
for  your  answer. 

7.  A  ball  weighing  2  kg.  is  suspended  from  a  cord  50  cm.  long,  and  made 
to  revolve  in  a  circle  whose  radius  is  30  cm.  Compute  (^d)  the  centripetal 
force  upon  the  ball  and  {b)  the  tension  upon  the  cord,  {c)  Draw  a  figure 
representing  the  conditions,  including  the  forces  involved. 


98  Dynamics 


VI.  Universal  Gravitation 

124.  Universal  Gravitation. — The  pupil  is  already  familiar 
with  the  fact  that  the  earth  attracts  all  bodies  at  and  near  its  sur- 
face, and  that  this  attraction  is  directly  evident  as  weight.  There 
is,  however,  no  familiar  evidence  that  all  bodies  attract  one  another 
under  all  circumstances,  yet  such  is  the  case.  The  attraction  be- 
tween masses  of  even  several  hundred  pounds  is  exceedingly 
small  —  so  small,  in  fact,  as  to  be  far  beyond  any  ordinary  means 
of  detecting  it.*  Nevertheless  a  number  of  experimenters  have 
measured  the  attraction  between  masses  of  various  substances 
varying  in  weight  from  a  fraction  of  an  ounce  to  several  hundred 
pounds.  These  experiments  not  only  prove  the  existence  of 
gravitational  attraction  between  bodies,  but  afford  a  measurement 
of  it  that  is  probably  not  in  error  by  more  than  one  per  cent. 
The  methods  by  which  such  exceedingly  delicate  measurements 
are  carried  out  lie  far  beyond  the  range  of  elementary  physics. 

Sir  Isaac  Newton  (1642-17 2 7),  a  noted  English  physicist  and 
mathematician,  proved  that  the  planets  are  held  in  their  orbits  by 
the  attraction  of  the  sun,  and  the  moon  in  its  orbit  by  the  attrac- 
tion of  the  earth ;  and  that  the  motions  of  the  planets  are  slightly 
modified  by  their  attractions  for  one  another.  He  also  discov- 
ered that  the  attractions  of  the  sun  and  the  planets  for  one 
another  and  the  attraction  of  the  earth  for  bodies  upon  its  surface 
are  all  in  accordance  with  the  same  law.  This  law  is  called 
Nev^'ton's  law  of  gravitation,  and  is  as  follows  :  — 

Every  particle  of  matter  in  the  universe  attracts  every  other 
particle  with  a  force  whose  direction  is  that  of  the  line  joining 
them,  and  whose  magnitude  is  directly  proportional  to  the  product 
of  their  masses,  and  inversely  proportional  to  the  square  of  the 
distance  between  them. 

1  Two  spheres  of  cast  iron  each  1.8  m.  in  diameter  would  attract  each  other  with 
a  force  of  i  g.  when  placed  close  together.  Such  spheres  would  weigh  about  22,000 
kg.  or  22  metric  tous  each. 


Universal  Gravitation  99 

Gravitation  is  the  general  term  applied  to  the  force  with  which 
all  bodies  attract  one  another.  The  attraction  of  the  earth  for 
bodies  at  and  near  its  surface  is  usually  called  gravity. 

125.  Illustrations  of  the  Law.  —  Newton  proved  that  the 
attraction  between  a  sphere  and  any  other  body  is  the  same  as  it 
would  be  if  the  entire  mass  of  the  sphere  were  concentrated  at  its 
center.  Hence  in  considering  the  attraction  of  the  earth  for  any 
body  upon  its  surface,  the  distance  stated  in  the  law  is  the  earth's 
radius.  In  considering  the  attraction  between  any  two  bodies  of 
appreciable  size,  the  distance  between  their  centers  of  gravity  is  to 
be  taken  as  the  distance  between  the  bodies. 

Let/ denote  the  attraction  between  two  masses  m^  and  m^,  and 
d  the  distance  between  their  centers  of  gravity  ;  similarly,  let  F 
denote  the  attraction  between  two  masses  J/j  and  Mc^,  and  D  the 
distance  between  their  centers  of  gravity ;  then,  according  to  the 

/./r..  __.___.  (14) 

The  meaning  of  the  law  as  expressed  in  this  proportion  will  be 
more  readily  understood  from  the  following  examples  :  — 

1.  What  is  the  relation  between  the  masses  of  two  bodies  and 
their  weights  ? 

Let  m\  and  M\  denote  the  masses  of  the  two  bodies,  and  /  and  F  their 
weights  respectively.  Since  the  second  attracting  body  in  each  case  is  the 
earth,  wo  =  M^  =  the  mass  of  the  earth  and  </  =  Z>  =  the  radius  of  the  earth. 
Hence  the  above  proportion  reduces  to 

/:F::mi  :  Mi; 

that  is,  the  weights  of  any  two  bodies  are  proportional  to  their  masses  —  a 
fact  with  which  the  pupil  is  already  familiar. 

2.  How  does  the  weight  of  a  body  upon  the  moon  compare 
with  the  weight  of  an  equal  mass  upon  the  earth  ? 

Let  mi  and  Afi  denote  the  equal  masses  upon  the  moon  and  the  earth,  and 
mo  and  Afo  the  masses  of  the  moon  and  the  earth  respectively.  Then  d  and 
D  denote  the  radii  of  the  moon  and  the  earth,  and /and  7^ the  weight  of  the 
body  upon  the  moon  and  upon  the  earth  respectively. 


loo  Dynamics 

Snce  mi  =  Mu  the  proportion  reduces  to 

from  which  /h-  /r=  ^  X  f  -  V- 

The  radius  of  the  earth  is  3960  mi.,  the  radius  of  the  moon  1082  mi.,  and 
the  mass  of  the  moon  ^  of  the  earth's  mass.  Substituting  these  values  in  the 
equation,  we  get 

f^F=i^x  (3960  +  io82)»  =  .1675  =  \  (nearly). 

3.  How  does  the  force  of  gravity  at  the  distance  of  the  moon 
compare  with  its  value  at  the  surface  of  the  earth? 

The  mean  distance  of  the  moon  is  almost  exactly  60  times  the  earth's 
radius.  Let  D  denote  the  radius  of  the  earth  ;  then  </,  the  distance  between 
the  centers  of  the  earth  and  the  moon,  etjuals  60  D.  Let  Afx  and  /;/i  denote 
equal  masses  upon  the  earth  and  at  the  distance  of  the  moon  respectively. 
J/j  =  wt  =  the  mass  of  the  earth.  F  is  the  force  of  gravity  upon  the  given 
mass  at  the  earth's  surface,  and /its  value  at  the  distance  of  the  moon.  The 
formula  then  becomes 

/    F        ^»^^g      A/iA/a 
-^         '  •  (60  Z?)«  '     D^    * 


from  which  /-*-  F= 


(60  Oy      6o5«     3600 


Thus  a  mass  of  3600  lb.  at  the  distance  of  the  moon  would  be  attracted  by 
the  earth  with  a  force  of  one  pound. 

Since  the  masses  involved  fn  this  problem  are  the  same  at  both  distances, 
a  simpler  and  more  direct  solution  is  obtained  by  making  use  only  of  the 
relation  that  the  force  varies  inversely  as  the  square  of  the  distance.  Thus 
the  ratio  of  the  distances  is  60,  the  square  of  this  ratio  is  36CX),  and  the  recipro- 
cal of  this  (taken  because  the  relation  is  inverse)  is  3^5. 

126.  Revolutioii  and  Rotation  of  the  Moon  and  the  Earth. — 
Since  gravity  at  the  distance  of  the  moon  is  ^^^  of  its  value  at 
the  earth's  surface,  the  acceleration  due  to  gravity  at  that  distance 
is  gg^QQ  of  32.15  ft.,  or  .1072  in.  per  sec.  per  sec.  A  body  at  the 
distance  of  the  moon,  starting  without  any  motion  in  the  direction 
of  the  earth,  would  fall  one  half  of  .1072  in.,  or  .0536  in.,  toward 
the  earth  during  the  first  second.     The  earth's  attraction,  there- 


Universal  Gravitation  loi 

fore,  acting  as  a  centripetal  force,  draws  the  moon  out  of  a  straight 
course  a  distance  of  .0536  in.  every  second.  As  the  moon's 
velocity  is  about  f  mi.  per  sec,  this  deflection  is  very  slight ;  but 
it  is  exactly  what  is  required  to  keep  the  moon  in  its  orbit. 

The  moon's  attraction  for  the  earth  is,  of  course,  equal  to  the 
earth's  attraction  for  the  moon ;  but  since  the  mass  of  the  earth 
is  80  times  that  of  the  moon,  the  effect  upon  the  earth's  motion  is 
proportionately  smaller.  The  earth  and  the  moon,  in  fact,  revolve 
in  the  same  direction  round  their  common  center  of  gravity ; 
which,  as  it  divides  the  distance  between  the  centers  of  the  two 
bodies  inversely  as  their  masses  (see  problem  9  following  Art.  84), 
lies  within  the  mass  of  the  earth  about  1 100  mi.  below  the  surface. 
(This  motion  of  the  earth  has  nothing  whatever  to  do  with  its 
rotation  on  its  axis.) 

The  sun's  attraction  deflects  the  earth  from  a  straight  course  by 
about  one  ninth  of  an  inch  in  a  second,  while  the  earth  is  traveling 
nearly  nineteen  miles.  The  mass  of  the  earth  is  so  great  that  the 
force  required  to  produce  even  so  slight  a  change  of  direction  is 
inconceivable,  being  no  less  than  3,600,000  millions  of  millions  of 
tons  (36  with  seventeen  ciphers). 

The  effect  of  the  sun's  attraction  for  the  planets  is  a  continuous 
change  of  direction  of  motion,  not  a  change  of  speed.  There  is 
no  force  acting  to  maintain  the  motion  of  the  planets,  and  none 
is  necessary,  since  the  heavenly  bodies  move  through  space  with- 
out friction  or  resistance  of  any  kind.  The  same  is  true  of  the 
rotation  of  the  sun  and  the  planets  upon  their  axes.  A  spinning 
top  is  brought  to  rest  by  the  resistance  of  the  air  and  the  friction 
upon  the  peg.  The  earth  rotates  without  friction,  hence  its  rate 
of  rotation  remains  constant  without  the  action  of  any  force  to 
maintain  it. 

127.  Effect  of  the  Earth's  Rotation  upon  Its  Shape.  —  If  the 
earth  were  fluid  (as  it  undoubtedly  once  was)  and  were  not  rotat- 
ing, the  gravitation  of  its  particles  would  cause  it  to  assume  the 
form  of  a  perfect  sphere.  The  rotation  of  a  fluid  planet  would 
cause  it  to  bulge  at  the  equator  and  flatten  at  the  poles,  until  the 


I02  Dynamics 

distortion  developed  a  centripetal  component  of  gravity  upon  each 
particle  sufficient  to  overcome  its  tendency  to  fly  off"  at  a  tangent. 
This  is  illustrated  in  Fig.  84,  which  repre- 
sents a  section  of  the  earth  taken  through 
the  axis  of  rotation  MN,  (The  departure 
from  the  spherical  shape  is  greatly  exagger- 
ated.) The  centripetal  force  upon  a  particle 
at  A  is  directed  toward  C,  the  center  of  the 
circle  described  by  A  about  the  axis.  This 
component  tends  to  draw  the  particle  toward 
the  pole,  but  is  just  sufficient  to  prevent 
it  from  moving  farther  toward  the  equator. 

The  earth  assumed  its  present  form  (disregarding  inequalities 
resulting  in  continents  and  oceans)  while  still  fluid ;  and,  as  a 
result  of  its  rotation,  the  polar  radius  is  nearly  13.5  mi.  less 
than  the  equatorial.  If  the  earth  were  to  stop  rotating,  the 
waters  of  the  ocean  would  flow  from  equatorial  regions  toward 
the  poles,  leaving  the  surface  of  the  ocean  truly  spherical.  The 
Mississippi  River  would  then  flow  toward  the  north,  for  its  mouth 
is  farther  from  the  center  of  the  earth  than  its  source. 

128.  Effect  of  the  Earth's  Shap^  and  Rotation  upon  Weight.  — 
If  a  body  were  carried  from  either  pole  toward  the  equator,  the 
earth's  attraction  upon  it  would  continually  decrease,  being  at  any 
latitude  inversely  proportional  to  the  square  of  the  earth's  radius 
at  that  latitude.  The  total  decrease  of  attraction  between  the 
pole  and  the  equator  is  about  -^  of  the  whole. 

All  bodies  on  the  earth  must  be  acted  upon  by  a  centripetal 
force  to  carry  them  round  with  the  earth  in  its  rotation.  A  certain 
portion  of  the  earth's  attraction  is  thus  employed  in  keeping 
bodies  from  flying  off",  and  only  the  remainder  of  this  attraction 
is  sensible  as  weight.  The  centripetal  force  increases  toward  the 
equator,  being  zero  at  the  poles  and  -,^\^  of  the  whole  attraction 
at  the  equator.  Since  the  centripetal  force  varies  as  the  square 
of  the  velocity  (Art.  120),  it  follows  that,  if  the  earth  rotated 
17    times   faster  than   it   does,  the    centripetal    force  would  be 


Universal  Gravitation 


103 


289  times  as  great  as  it  is  (17-  =  289),  and  bodies  at  the  equator 
would  weigh  nothing ;  i.e.  they  would  require  no  sustaining 
force  to  keep  them  from  falling,  for  the  earth's  attraction  would 
be  just  sufficient  to  keep  them  from  flying  off  at  a  tangent  as  mud 
flies  from  the  wheels  of  a  carriage. 

These,  therefore,  are  the  causes  of  the  difference  between 
weight  and  the  earth's  attraction  referred  to  in  the  note  to 
Art.  10.  As  the  result  of  the  earth's  shape  and  rotation  com- 
bined, the  weight  of  a  body  would  be  very  nearly  ^Jr  ^^^s  at  the 
equator  than  at  either  pole.  Thus  a  body  weighing  191  lb.  at 
the  pole  would  weigh  only  190  lb.  at  the  equator. 

129.  Cause  of  Gravitation.  —  The  cause  of  gravitation  is  not 
known,  neither  is  there  any  generally  accepted  theory  as  to  its 
cause.  It  acts  without  visible  or  material  connection  between  the 
attracting  bodies ;  yet  we  must  suppose  that  there  is  something 
pervading  all  space  by  means  of  which  and  through  which  it  is 
exerted.  It  is  inconceivable  that  two  bodies  not  in  contact 
should  be  able  to  act  upon  each  other  with  absolutely  nothing 
between  them.  Since  gravitation  acts  undiminished  in  a  vacuum 
and  beyond  the  limits  of  the  atmosphere,  it  is  clear  that  the 
means,  or  medium^  for  the  transmission  of  gravitational  force  is 
not  a  solid,  a  liquid,  or  a  gas,  an^  hence  is  not  matter  in  any  of 
its  ordinary  forms. 

PROBLEMS 

1.  («)  Would  the  variation  of  weight  at  different  latitudes  be  indicated  by 
any  form  of  balance  by  which  the  object  weighed  is  balanced  by  "  weights  "  ? 
(J))  Would  it  be  indicated  by  an  accurate  spring  balance  ?  Give  reasons  for 
each  answer. 

2.  What  fraction  of  its  weight  would  an  object  lose  when  taken  from  sea 
level  to  a  height  of  4  mi.? 

3.  The  diameter  of  Mars  is  4230  mi.  and  its  mass  is  approximately  one- 
ninth  of  the  earth's  mass.  How  does  gravity  upon  its  surface  compare  with 
gravity  upon  the  earth  ? 

4.  What  is  the  acceleration  of  a  falling  body  upon  Mars  ? 

5.  Why  does  the  atmosphere  not  offer  resistance  to  the  rotation  or  to  the 
revolution  of  the  earth? 


1 04  Dynamics 


6.  Is  the  acceleration  of  a  falling  body  due  to  the  whole  of  the  earth's 
attraction  or  to  the  part  that  we  call  weight  ? 

7.  What  would  be  the  subsequent  motion  of  the  moon  and  the  planets  if 
gravitation  should  suddenly  cease  to  act  upon  them  ? 

8.  The  average  specific  gravity  of  the  whole  earth  is  about  5.56.  (a)  How 
would  gravity  compare  with  its  present  value  if  the  average  density  of 
the  earth  were  equal  to  the  density  of  water  ?  (fi)  What  would  be  the 
acceleration  of  a  falling  body  in  that  case  ? 


VII.    The  Pendulum 

130.  Simple  and  Compound  Pendulums.  —  We  have  seen  that, 
when  a  suspended  body  is  drawn  asitle  from  a  position  of  stable 
equilibrium  and  released,  its  weight  causes  it  to  swing  to  and  fro 
about  the  position  of  equilibrium  until  it  is  brought  to  rest  by  fric- 
tion (Art.  75 ) .  Any  body  suspended  thus  and  free  to  swing  to  and 
fro,  ox  vibrate t  is  called  di  pendulum.  The  best  form  of  pendulum 
for  experimental  work  consists  of  a  small  sphere  of  some  dense 
material,  suspended  from  a  fixed  support  by  a  slender  thread. 
The  sphere  is  usually  called  the  bob  of  the  pendulum.  The  length 
of  such  a  pendulum  is  taken  as  the  distance  from  the  point  of  sus- 
pension to  the  center  of  the  bob.  This  is  not  perfectly  correct, 
but  there  is  no  appreciable  error  if  the  length  of  the  thread  is  not 
less  than  six  times  the  diameter  of  the  bob.  A  pendulum  of  this 
description  is  usually  called  a  simple  petidulum  ;  although,  strictly 
speaking,  a  simple  pendulum  consists  of  a  heavy  particle  having 
no  appreciable  size,  and  suspended  by  a  line  without  mass.  This 
is  purely  a  mathematical  conception,  useful  in  the  mathematical 
study  of  pendular  motion. 

Any  pendulum  having  an  appreciable  portion  of  its  mass  else- 
where than  in  a  compact  bob  at  the  end  is  called  a  compound 
pendulum.  Pendulums  for  other  than  experimental  purposes  are 
always  compound. 

131.  The  Motion  of  a  Pendulum.  —  After  a  pendulum  has  been 
drawn  aside  and  released,  the  bob  is  under  the  action  of  its  weight 
and  the  tension  of  the  thread  (friction  being  disregarded) .     The 


The  Pendulum 


105 


Fig.  85. 


tension  acting  always  at  right  angles  to  the  path  of  the  bob,  causes 
a  continuous  change  of  direction  of  the  bob,  but  does  not  affect 
its  speed.  The  weight  of  the  bob 
may  be  resolved  into  two  components, 
/  and /(Fig.  85),  at  any  point  of  the 
path.  The  component/,  taken  always 
at  right  angles  to  the  path,  is  balanced 
by  a  part  of  the  tension  of  the  thread  ; 
the  component/,  taken  along  the  tan- 
gent, acts  in  the  direction  of  motion 
while  the  bob  is  descending,  and  op- 
posite to  the  direction  of  motion  while 
it  is  rising.  The  motion  is  therefore  p 
accelerated  to  the  lowest  point  and 
retarded  from  that  point  to  the  end 
of  the  swing. 

It  is  evident  that  the  tangential  force  /  decreases  toward  the 
lowest  point  of  the  path,  where  it  is  zero ;  and  that  it  has  equal 
values  at  equal  distances  on  the  two  sides  of  this  point.  Hence 
the  bob  would  be  in  equilibrium  in  its  lowest  position,  if  at  rest ; 
but,  if  in  motion,  it  would  not  be  stopped  by  the  action  of  its 
weight  alone  until  it  had  risen  through  an  arc  equal  to  that 
through  which  it  had  fallen.  If  it  were  not  for  friction,  therefore, 
a  pendulum,  when  once  started,  would  vibrate  indefinitely.  It  is 
brought  to  rest  by  the  friction  of  the  air  and  the  friction  at  the 
point  of  support. 

132.  Definitions.  —  A  complete  swing  of  a  pendulum  in  one 
direction  is  called  a  vibration.  The  time  occupied  in  making  a 
vibration  is  called  the  time  of  vibration  or  period.  The  period  of 
a  pendulum  is  always  measured  in  seconds.  The  rate  of  a  pendu- 
lum is  the  number  of  vibrations  that  it  makes  in  one  second. 
The  angle  between  the  vertical  and  the  direction  of  a  pendulum 
at  the  end  of  a  vibration  is  called  the  amplitude  (angle  AOM, 
Fig.  85).  The  length  of  a  pendulum  consisting  of  a  small  bob  sus- 
pended by  a  thread  is  (very  approximately)  the  distance  from  the 


1 06  Dynamics 

point  of  suspension  to  the  center  of  the  bob.  The  length  of  a 
compound  pendulum  is  defined  as  the  length  of  a  simple  pendu- 
lum having  the  same  period.  (It  will  be  found  by  trial  that  this 
is  greater  than  the  distance  from  the  point  of  suspension  to  the 
center  of  gravity  of  the  pendulum  and  less  than  the  length  of  the 
body.) 

Laboratory  Exercise  22, 

133.  Effect  of  Amplitude  on  the  Period  of  a  Pendulum.  —  If 
two  pendulums  of  exactly  equal  length,  having  bobs  of  the  same 
size  and  material,  are  started  together  with  unequal  amplitudes, 
any  difference  in  their  periods  will  evidently  be  due  to  the  differ- 
ence in  their  amplitudes.*  Experiment  shows  that  the  periods  are 
equal  if  the  greater  amplitude  does  not  exceed  5°,  but  that  the 
difference,  though  small,  is  readily  appreciable  if  the  greater  ampli- 
tude is  above  20°.  The  period  of  a  pendulum  is  constant  for  am- 
plitudes less  than  j*  /  for  larger  amplitudes  the  period  increases 
very  slightly  with  increase  of  amplitude. 

It  follows  that,  as  the  amplitude  decreases,  the  average  speed  of 
the  bob  decreases  proportionally  (for  small  amplitudes)  ;  other- 
wise the  period  would  not  remain  constant.  The  decrease  of 
speed  is  due  to  the  fact  that,  as  the  arc  through  which  the  bob 
swings  grows  less,  the  average  value  of  the  tangential  force  / 
(Fig.  85)  also  grows  less. 

134.  Effect  of  Mass  and  Material.  —  The  effect  of  the  mass  or 
the  material  of  a  pendulum  is  investigated  by  starting  together  two 
pendulums  of  equal  length,  having  bobs  of  unequal  mass  or  of 
different  material,  or  both.  Neither  gains  on  the  other ;  hence 
the  period  is  independent  of  the  mass  and  also  of  the  material  of 
the  pendulum.  The  reason  for  this  is  similar  to  that  for  the  equal 
acceleration  of  falling  bodies  (Arts.  loi  and  112),  —  the  tangential 


1  If  two  pendulums  are  started  simultaneously,  even  a  very  slight  difference  in 
their  periods  will  be  evident  from  the  fact  that,  after  swinging  for  a  minute  or  less, 
the  ptendulum  having  the  shorter  period  will  reach  the  extremity  of  its  arc  in  ad- 
vance of  the  other.  This  test  is  at  least  ten  times  as  delicate  as  that  of  counting 
the  vibrations  of  ^ ach  pendulum  for  one  minute. 


The  Pendulum 


107 


component  of  the  weight  of  a  pendulum  bob  is  proportional  to  its 
mass. 

135.  Effect  of  Length  on  the  Period.  —  If  the  length,  /,  and  the 
period,  /,  of  different  pendu]ums  are  accurately  determined,  it  will 
be  found  that  the  ratio  V/ :  /  has  the  same  value  for  all.  That 
is,  letting  /j  and  4  denote  the  Jengths  of  any  two  pendulums  and 
A  and  /j  their  periods,  then  V4:  A  : :  V^:  4,  or,  taking  the  ratios 
of  like  quantities,  which  is  preferable. 


From  which,  by  squaring  the  ratios, 

/2  .   /2  . .  /    .   / 


(15) 


(16) 

Stated  in  words,  the  period  of  a  pendulum  is  proportional  to  the 
square  root  of  its  length  ;  or,  the 
square  of  the  period  is  propor- 
tional to  the  length. 

Figure  86  shows  why  the  period 
increases  with  the  length.  The 
tangential  force  /  acting  on  the 
bob  of  the  pendulum  OA  is  much 
less  than  that  upon  the  bob  of  the 
pendulum  O^A  at  an  equal  dis- 
tance from  the  vertical.  Hence 
the  acceleration  of  the  longer 
pendulum  is  less  and  its  period 
greater  than  that  of  the  shorter 
pendulum.  (This,  of  coarse,  does 
not  establish  the  definite  relation 
stated  in  the  law.) 

136.  Effect  of  Gravity.  —  When  a  pendulum  is  taken  from  one 
location  to  another  in  which  the  force  of  gravity  is  different,  its 
rate  is  affected  in  the  same  manner  as  that  of  a  falling  body,  and 
for  the  same  reasons  (Arts.  98  and  1 13).  If  the  force  of  gravity  is 
less  in  the  new  location,  the  tangential  component  of  the  weight 


Fig.  86. 


io8  Dynamics 

of  the  bob  will  be  proportionately  less  and  the  period  will  be 
increased.  The  period  of  a  pendulum  is  inversely  proportional  to 
tht  square  root  of  the  acceleration  of  a  falling  body.  Stated  alge- 
braically, the  law  is 

tx'.t^w  -<fgi\  Vgi.  (17) 

The  effect  of  an  increase  in  the  force  of  gravity  is  illustrated  by 
means  of  a  pendulum  with  an  iron  bob.  By  holding  an  end  of  a 
strong  bar  magnet  under  and  near  the  bob,  its  motion  will  be  con- 
trolled by  its  weight  and  the  attraction  of  the  magnet  acting  to- 
gether, the  latter  force  being  equivalent  to  an  increase  of  gravity. 
With  a  small  amplitude  of  vibration,  the  bob  does  not  swing 
beyond  the  strong  attraction  of  the  magnet  and  the  period  is  con- 
siderably shortened  {Exp.). 

137.  The  Pendulum  Formula.  —  It  can  be  shown  by  mathe- 
matical analysis  based  upon  the  second  law  of  motion  that  the 
period  of  a  pendulum,  provided  the  amplitude  be  small,  is  given 
by  the  formula 


'=4i-> 


(.8) 

in  which  /  denotes  the  period,  and  /  the  length  of  the  pendulum, 
g  the  acceleration  of  a  falling  body,  and  tt  the  ratio  of  the  circum- 
ference of  a  circle  to  its  diameter  (=3.1416). 

The  formula  includes  the  four  laws  of  the  pendulum  already 
considered.  The  first  two  laws  follow  from  the  fact  that  the 
formula  contains  no  factor  depending  upon  amplitude,  mass,  or 
material.  That  the  period  is  proportional  to  the  square  root  of  / 
and  inversely  proportional  to  the  square  root  of  g  is  evident  from 
the  formula  as  it  stands.  The  formula  holds  for  the  compound 
pendulum  provided  the  length  be  taken  as  defined  in  Art.  132. 

138.  Uses  of  the  Pendulum.  —  The  principal  use  of  the 
pendulum  is  to  regulate  the  motion  of  clocks.  The  mechanism  by 
which  this  is  effected  is  represented  in  Fig.  87.  The  pendulum 
rod  passing  between  the  prongs  of  a  fork,  A^  communicates  its 
motion  to  a  rod,  B,  which  turns  on  a  horizontal  axis,  C.     To  this 


The  Pendulum 


109 


axis  is  fixed  a  curved  piece,  called  the  escapement,  which  has  a 
projection  at  each  end.  These  projections  are  alternately  brought 
into  contact  with  the  teeth  of  the  escape7nent 
wheel,  D,  by  the  motion  of  the  pendulum.  The 
escapement  wheel  is  the  last  of  the  train  of 
wheels  in  the  clock  and  is  driven  by  them ; 
but  the  escapement  permits  only  one  tooth  to 
pass  at  a  time,  while  the  pendulum  swings  back 
and  forth.  The  control  thus  exercised  on  the 
escapement  wheel  by  the  pendulum  is  com- 
municated through  the  entire  train  of  wheels  to 
the  weight  or  spring  that  runs  the  clock  and  to 
the  hands.  As  the  escapement  wheel  turns,  its 
teeth  press  upon  the  projections  of  the  escape- 
ment. These  slight  impulses  are  transmitted  to 
the  pendulum  and  maintain  its  motion. 

The  rate  of  a  clock  is  controlled  by  means  of 
a  thread  and  nut  at  the  lower  end  of  the  pendu- 
lum. The  bob  is  raised  or  lowered  by  turning 
this  nut. 

A  compound  pendulum  of  special  construc- 
tion is  used  to  determine  the  acceleration  due 
to  gravity  at  different  places.  The  length  and 
period  of  the  pendulum  are  determined  very 
accurately,  and  their  values  substituted  in  the  pendulum  formula, 
from  which  the  value  of  ^  is  then  computed. 

PROBLEMS 

•  I.  How  would  the  expansion  of  the  rod  of  a  pendulum  in  summer  and  its 
contraction  in  winter  affect  the  rate  of  a  clock  if  the  height  of  the  bob  were 
not  adjusted  to  compensate  the  expansion  and  contraction  ? 

2.  What  is  the  usual  shape  of  the  bob  of  a  clock  pendulum  ?     What  is  the 
advantage  of  this  shape  ? 

3.  What  is  the  length  of  a  pendulum  that  beats  seconds  (/  =  i)  at  a  place 
where  the  value  of  g  is  980  cm? 

SuoGESTioN.— Substitute  the  values  of  /  and  g  in  the  pendulum  formula, 
and  solve  for  /. 


Fig.  87. 


iio  Dynamics 


4.  Find  the  lengths  of  the  pendulums  whose  periods  are  .7  sec.  and  1.5 
sec  respectively. 

Suggestion.  —  Substitute  in  the  pendulum  formula,  assuming  980  cm.  for 
g ;  or  substitute  in  formula  16,  taking  /^  =  i,  for  /j  the  length  of  the  seconds 
pendulum,  and  for  t\  the  given  period.     Solve  for  l\. 

5.  find  the  periods  of  pendulums  whose  lengths  are  20  cm.  and  250  cm. 
respectively. 

6.  What  is  the  length  of  a  pendulum  that  makes  70  vibrations  per  minute? 

7.  What  is  the  length  of  the  seconds  pendulum  on  Mars? 

8.  At  what  point  or  points  in  the  path  of  a  pendulum  bob  is  its  speed 
greatest  ?  least  ?  increasing  most  rapidly  ?  constant  ?  Is  the  speed  constant 
for  any  appreciable  distance  ?  Is  the  increase  of  speed  constant  in  any  part 
of  the  path  ? 

Si'GGESTiov. — The  answers  are  all  to  be  determined  from  a  knowledge  of 
the  tangent^  force  at  any  point  of  the  path. 


CHAPTER  VI 

ENERGY 

I.    Energy  and  Work 

139.  Energy. — The  word  energy  is  used  in  science  with  a 
definite  meaning  which  i:an  be  understood  only  by  a  study  of  the 
different  forms  in  which  energy  exists.  Energy  is  most  generally 
recognized  by  the  ability  of  bodies  possessing  it  to  cause  motion 
in  other  bodies  or  to  maintain  their  motion  in  opposition  to  friction 
or  other  forces  tending  to  stop  them.  Thus  the  energy  of  a  bent 
bow  is  shown  by  its  ability  to  project  an  arrow,  and  the  energy  of 
a  coiled  spring  by  its  ability  to  run  a  clock.  The  energy  of  the 
wind  enables  it  to  turn  windmills,  propel  ships,  uproot  trees,  etc. 
The  energy  of  coal,  wood,  and  oil  is  utilized  by  means  of  the 
steam  engine  in  running  mills,  drawing  trains,  and  propelling 
steamships.  Energy,  in  fact,  is  manifested  in  one  or  more  of  its 
many  forms  in  every  phenomenon. 

140.  Kinetic  Energy.  —  A  moving  body  can  impart  motion  to 
other  bodies ;  it  therefore  has  energy.  The  energy  that  a  body 
has  by  virtue  of  its  mass  and  its  velocity  is  called  energy  of  motion 
or  kinetic  energy, 

141.  Work.  —  The  transference  of  energy  from  one  body  to 
another  is  called  work.  The  amount  of  energy  transferred  and 
the  amount  of  work  done  are  equivalent  expressions.  Energy  is 
transferred  from  one  body  to  another  by  means  of  the  force  that 
each  exerts  upon  the  other.  For  example,  when  one  ball  rolls 
against  another,  setting  it  in  motion,  the  latter  receives  kinetic 
energy  by  means  of  the  pressure  exerted  upon  it  while  the  two 
are  in  contact ;  and,  at  the  same  time,  the  energy  of  the  first  ball 


1 1 2  Energy 

is  diminished  by  an  equal  amount  by  the  reaction  of  the  second 
ball  upon  it  in  the  direction  opposite  to  its  motion.  The  first  ball 
is  said  to  do  positive  work  upon  the  second,  and  the  latter  negative 
work  upon  the  first.  The  work  by  which  a  body  gains  energy 
is  called  positive,  and  that  by  which  it  loses  energy  is  called 
negative. 

To  illustrate  further :  Positive  work  is  done  upon  a  train  by  the 
pull  of  the  engine.  This,  if  it  were  the  only  force  acting  on  the 
train,  would  increase  its  kinetic  energy ;  but  the  retarding  forces 
of  friction  are  at  the  same  time  doing  negative  work  upon  the 
train.  If  the  pull  exceeds  friction,  positive  work  will  exceed  nega- 
tive, and  the  excess  of  positive  work  wjll  result  in  increase  of 
kinetic  energy.  If  the  two  are  equal  (resultant  force  zero),  the 
positive  and  the  negative  work  will  be  equal  and  the  energy  of  the 
train  will  be  constant  (the  speed  being  constant) .  After  the  engine 
ceases  to  pull,  the  only  work  done  upon  the  train  (provided  the 
track  be  level)  is  the  negative  work  due  to  friction,  by  which  the 
train  loses  its  energy  and  is  stopped. 

142.  Conditions  Necessary  for  the  Transference  of  Energy. — 
Since  one  body  does  work  upon  another  only  by  exerting  force  upon 
it,  we  commonly  say  that  the  force  does  the  work.  Although  work 
is  done  only  through  the  action  of  force,  a  force  may  act  without 
doing  work.  When  a  horse  pulls  on  a  load  without  starting  it,  no 
work  is  done  by  the  force  exerted,  for  there  is  no  transference  of 
energy  to  the  load.  If  the  load  is  started,  the  force  acts  through 
a  certain  distance,  and,  in  doing  so,  does  work  of  acceleration  in 
imparting  motion  to  the  load  and  work  against  resistance  in  over- 
coming friction.^  There  is  neither  work  of  acceleration  nor  work 
against  resistance  without  motion  of  the  body  upon  which  the  force 
acts. 

A*  force  may  also  act  upon  a  moving  body  without  doing  work. 
Whether  work  will  be  done    or  not  depends  upon  the  relative 

1  Fricrion  develops  heat,  and  heat  is  a  form  of  energy  (Art.  148) ;  hence  to 
maintain  motion  against  friction  involves  the  transference  of  energy,  or  the  doing 
of  work. 


Energy  and  Work  1 1 3 

direction  of  the  force  and  the  motion,  as  will  be  seen  from  the 
following  illustrations.  The  weight  of  a  projectile  rising  vertically 
constantly  diminishes  its  kinetic  energy  by  doing  negative  work  of 
acceleration.  The  weight  of  a  falling  body  does  positive  work  of 
acceleration  upon  it,  constantly  increasing  its  kinetic  energy.  In 
both  cases  the  line  of  action  of  the  force  (weight)  \^  parallel  to 
the  direction  of  motion.  As  a  pendulum  vibrates,  it  is  only  the 
tangential  component  of  the  weight  of  its  bob  that  does  work  of 
acceleration  (Fig.  85),  causing  gain  of  kinetic  energy  during  the 
first  half  of  each  vibration  and  loss  of  kinetic  energy  during  the 
second  half;  and  it  is  only  this  component  that  does  work  against 
the  resistance  of  the  air.  The  other  component  of  weight,  which 
acts  at  right  angles  to  the  direction  of  motion,  does  no  work,  for 
it  neither  changes  the  speed  of  the  bob  nor  helps  to  maintain  its 
motion  against  the  friction  of  the  air.  The  same  is  true  of  the 
tension  of  the  thread. 

A  force  acting  upon  a  body  at  right  angles  to  its  direction  of 
motion  does  no  work  ;  for  it  neither 
changes  the  speed  of  the  body  nor 
helps  to  maintain  its  motion  against 
friction  or  other  resistance.  This 
is  true  whether  the  force  is  bal- 
anced or  unbalanced.  Thus,  on 
the  whole,^  no  work  is  done  upon 
the  moon  by  the  attraction  of  the 

earth,  or  upon  the  planets  by  the 

Fig.  80. 
attraction  of  the  sun.     In  the  case 

of  oblique  forces,  as  the  weight  of  a  pendulum  bob,  it  is  only  the 
tangential  component  of  the  force  that  does  work  ;  that  is,  tiie  com- 
ponent whose  line  of  action  is  parallel  to  the  direction  of  motion. 

1  Since  the  orbits  of  the  moon  and  the  earth  are  not  quite  circular  but  elliptical, 
the  attraction  is  not  exactly  at  right  angles  to  the  path  except  at  two  points  {A  and 
B,  Fig.  88).  As  the  planet  movt-s  from  A  to'B,  the  tangential  component  of  the 
attraction  causes  decrease  of  speed,  while  from  i?  to  ^  it  causes  an  equal  increase. 
Hence,  on  the  whole,  there  is  neither  gain  nor  loss  of  kinetic  energy.  There  is,  of 
course,  no  work  against  resistance,  for  the  planets  move  without  friction. 


114 


Energy 


Fig.  89. 


143.  Measure  of  Work.  —  The  work  done  by  a  force  is  meas- 
ured by  the  product  of  the  force  and  the  distance  through  which 
the  force  acts,  the  distance  being  always  measured  parallel  to  the 
bne  of  action  of  the  force. 

For  example,  suppose  we  wish  to  find  the  work  done  by  the 
weight  of  a  ball  while  it  rolls  down  an  in- 
chned  plane,  AB  (Fig.  89).  Let  ^denote  the 
length  of  the  plane,  AB^  and  h  the  height  of 
the  plane,  BC.  The  part  of  w  that  does  work 
is  the  component,  /,  acting  parallel  to  the 
plane.  The  distance  through  which  it  acts  is 
//,  and  the  work  that  it  does  is  measured  by  df. 
From  similar  triangles,  f  \  w  \\  h  \  d  )  hence 
■"c  fd  =  wh.  That  is,  the  work  done  by  the 
weight  of  the  ball  is  also  measured  by  wh. 
But,  while  the  ball  rolls  down  the  plane  the  distance  d^  it  descends 
through  the  vertical  distance  h.  Thus  the  work  done  by  w  is 
equal  to  the  product  of  w  and  the  displacement  in  its  own  direc- 
tion. 

In  measuring  the  work  done  by  an  oblique  force,  it  is  immate- 
rial whether  we  take  ( i )  the  product  of  the  whole  force  and  the 
component  of  the  displacement  taken  parallel  to  the  line  of  action 
of  the  force  {wh  in  the  above  example),  or  (2)  the  product  of  the 
whole  displacement  and  the  component  of  the  force  acting  parallel 
to  it  {fd  in  the  above  example).  Both  methods  are  fully  included 
in  the  above  rule  for  the  measure  of  work.  The  choice  is  a  matter 
of  convenience,  determined  by  the  nature  of  the  problem.  In  the 
case  where  the  applied  force  and  the  displacement  have  the  same 
or  opposite  directions,  the  work  done  is  simply  the  product  of  the 
two.  If  the  force  is  variable,  the  average  force  exerted  through 
the  distance  considered  must  be  taken.  The  work  done  by  a 
force,  /,  while  acting  through  a  distance,  </,  in  its  own  line  of 
direction^  is  expressed  by  the  formula 


Work=fd. 


(19) 


Energy  and  Work 


<i5 


144.  A  Further  Illustration.  —  The  fact  that  the  work  done  by 
a  force  is  equal  to  the  product  of  the  force  and  the  displacement 
in  its  own  line  of  action,  regardless  of  the  actual  path  of  the  body, 
is  beautifully  illustrated  by  the  following  experiment.  A  heavy 
lead  or  iron  ball  is  suspended  by  a  cord  1.5  m.  to  2  m.  long  so  that 
it  will  swing  as  a  pendulum  in  front  of  a  blackboard  and  within 
a  few  centimeters  of  it ;  and  parallel  horizontal  lines  are  drawn 
on  the  board  a  short  distance  apart  to  mark  elevations.  The  pen- 
dulum is  started  at  an  angle  of  about  30°,  with  the  bob  at  the 
height  of  one  of  the  lines  on  the  board.  It  swings  to  (about)  the 
same  height  on  the  other  side  (the  effect  of  the   resistance  of 


the  air  being  nearly  or  quite  inappreciable  for  a  single  swing).  In 
Fig.  90  BCD  represents  the  path  of  the  bob.  A  nail  or  a  rod  is  now 
driven  into  the  blackboard  at  A,  vertically  below  the  support  of  the 
pendulum,  and  a  little  above  the  line  BD ;  and  the  pendulum  is 
started  from  the  same  height  as  before.  When  it  reaches  the  ver- 
tical position,  the  cord  is  caught  by  the  nail  at  A,  and  the  bob 
rises  through  the  arc  CE,  having  ^  as  a  center.  The  point  E  to 
which  the  bob  rises  has  the  same  elevation  as  B  and  D,  and  the 
bob  again  rises  to  B  in  its  return.  This  shows  that  the  bob  loses 
equal  kinetic  energy  in  rising  the  same  vertical  distance  by  the  two 


1 1 6  Energy 

paths  CD  and  CEy  and  gains  equal  kinetic  energy  in  descending 
the  same  vertical  distance  by  the  two  paths.  Since  the  gain  or 
loss  of  kinetic  energy  is  equal  to  the  work  done  by  the  weight  of 
the  bob  (the  tension  of  the  cord  does  no  work  in  any  case),  it 
follows  that  the  work  done  is  determined  by  the  vertical  displace- 
ment and  is  independent  of  the  path.  If  w  is  the  weight  of  the 
bob  and  h  the  height  of  B  above  C,  the  work  done  in  either  half 
of  the  vibration  is  wh.  The  bob  has  at  C  the  same  kinetic 
energy  and  hence  the  same  velocity  as  it  would  have  if  it  fell  ver- 
tically through  the  distance  h. 

146.  Units  of  Work.  —  There  are  several  units  of  work.  We 
shall  define  and  use  only  three,  liht  f oof-pound  (ft. -lb.)  is  the 
work  done  when  a  force  of  one  pound  acts  through  a  distance  of 
one  foot.  The  kilogram- meter  (kg.-m.)  is  the  work  done  when  a 
force  of  one  kilogram  acts  through  a  distance  of  one  meter.  The 
gram-centimeter  (g.-cm.)  is  the  work  done  when  a  force  of  one 
gram  acts  through  a  distance  of  one  centimeter. 

Examples.  —  i.  The  pressure  exerted  by  the  expanding  gases  behind  a 
cannon  ball  when  it  is  fired,  is  exerted  from  the  breech  to  the  mouth  of  the 
cannon.  If  this  distance  is  12  ft.  and  the  average  total  pressure  upon  the 
ball  is  40,000  lb.,  the  work  done  upon  the  ball  is  40,cxx)  x  12,  or  480,000 
ft.-lb.  Hence,  disregarding  the  comparatively  small  amount  of  work  done 
against  friction,  the  ball  leaves  the  cannon  with  480,000  ft.-lb  of  kinetic 
energy. 

2.  If  a  mass  of  7  kg.  falls  a  distance  of  30  m.,  its  weight  does  7  x  30,  or 
210,  kg.-m.  of  work  upon  it ;  and  this  is  also  the  measure  of  the  kinetic  energy 
of  the  body  at  the  end  of  its  fall. 

146.  Measure  of  Kinetic  Energy. — The  kinetic  energy  of  a  body 
is  equal  to  the  work  that  would  be  done  by  any  unbalanced  force 
in  imparting  the  given  velocity  to  the  body.  The  formula  for 
kinetic  energy  is  most  simply  derived,  however,  by  assuming  that 
the  given  velocity  is  imparted  to  the  body  by  its  weight  in  falling. 
In  this  case  the  force,/,  is  numerically  equal  to  the  mass, ;«,  of  the 
body.  Let  d  denote  the  distance  the  body  must  fall  to  acquire  the 
given  velocity,  v,  starting  from  rest,  and  g  the  acceleration. 

The  kinetic  energy  {K.E.)  that  would  be  imparted  to  the  body 


Energy  and  Work  117 

while  falling  is  measured  by  the  work  that  would  be  done  upon 
the  body  by  its  weight  ;  that  is, 


K.E.=fd. 


Since 

d  = 

I' 
'  2g 

and 

/= 

m, 

we  have, 

by 

substitution. 

K.E.= 

fmP' 

^S 


(equation  10,  Art.  97) 


(20) 


That  is,  the  kinetic  energy  of  a  body  (whether  a  falling  body  or 
any  other)  is  measured  by  the  product  of  its  mass  and  the  square 
of  its  velocity,  divided  by  twice  the  acceleration  of  a  falling  body. 
Kinetic  energy  is  measured  in  foot-pounds,  kilogram-meters,  or 
gram-centi:neters  according  as  the  units  of  mass  and  distance 
employed  are  the  pound  and  the  foot,  the  kilogram  and  the 
meter,  or  the  gram  and  the  centimeter.  In  coming  to  rest,  a 
body  will  do  work  equal  to  its  kinetic  energy. 

It  is  mass  and  not  weight  that  is  involved  in  kinetic  energy. 
The  kinetic  energy  of  a  body  moving  with  a  given  velocity  would 
be  the  same  in  regions  so  far  from  the  earth  or  any  other  heavenly 
body  that  it  had  no  weight  at  all. 

Examples. —  i.  Find  the  kinetic  energy  of  a  car  weighing  12  tons,  and 
moving  with  a  velocity  of  40  ft.  per  sec. 


„  _       24000  X  402 

K.E.  —  — —  597,015  ft.-lb. 

2x32.16       ^^'      ^ 


2.  If  the  velocity  of  the  car  is  imparted  by  a  constant  pull  of  800  lb.  and 
the  friction  is  150  lb.,  through  what  distance  does  the  force  act? 

One  hundred  and  fifty  pounds  of  the  pull  does  work  against  friction  ; 
only  the  remainder  acts  as  an  unl)alanced  force  to  cause  acceleration  ;  hence 
650  d=  597,015.     From  which  d  =  918.5  ft. 

3.  How  far  would  the  car  run  before  being  brought  to  rest  by  friction 
acting  alone? 

Since  friction  must  do  597,015  ft.-lb.  of  negative  work  upon  the  car,  the 
distance  d  is  found  from  the  equation  150  d=  597»oi5« 


1 1 8  Energy 

147.  Potential  Energy.  —  In  bending  a  bow,  work  is  done  in 
opposition  to  its  elasticity,  —  a  force  whicii  not  only  opposes  the 
distortion,  but  also  tends  to  restore  the  original  shape  of  the  body 
by  bringing  its  parts  into  their  former  relative  positions.  The 
bow,  when  released,  is  therefore  able  to  exert  a  force  through  a 
certain  distance,  or,  in  other  words,  it  can  do  work.  This  is  what 
we  have  in  mind  when  we  say  that  a  bent  bow  possesses  energy. 
It  has  the  ability  to  cause  motion  in  itself  or  otiier  bodies,  although 
it  is  not  actually  in  motion.  Energy  existing  in  this  form  is  called 
potential  fntr^'.  The  energy  of  the  bent  bow  is  evidently  the 
result  of  the  work  done  in  bending  it. 

Other  examples  of  bodies  having  potential  energy  are  the  coiled 
spring  of  a  watch,  a  raised  clock  weight,  the  compressed  air  in  an 
air  rifle,  and  the  lifted  ram  or  hammer  of  a  pile  driver.  The 
coiled  spring,  like  the  bow,  exerts  a  force  in  recovering  from  dis- 
tortion, and  the  compressed  air  in  expanding ;  in  both  cases  there 
is  relative  motion  of  the  parts  of  the  body.  The  clock  weight 
exerts  force,  and  hence  does  work,  throughout  its  descent  ;  the 
hammer  of  a  pile  driver  does  work  only  at  the  end  of  its  fall.  In 
either  case  the  ability  of  the  body  to  do  work  is  due  to  the  fact 
that  its  weight  does  work  upon  //  during  its  descent.  The  poten- 
tial energy  of  a  body  at  any  elevation  is  therefore  measured  by 
the  product  of  its  weight  and  the  elevation,  and  is  independent 
of  the  path  by  which  the  body  descends  to  the  lower  level  and 
also  of  the  time  occupied  in  making  the  descent  (Art.  144). 

The  energy  that  a  body  has  by  virtue  of  its  position  or  the 
relative  position  of  its  parts  is  C2i\\^d  potential  energy.  This  defini- 
tion implies  the  existence  of  force  tending  to  move  the  body  or 
to  cause  relative  motion  of  its  parts,  and  also  the  existence  of  room 
for  such  motion  to  take  place. 

148.  Heat  is  a  Form  of  Energy.  —  Through  our  common  ex- 
perience we  are  familiar  with  many  effects  of  heat  ;  yet  these 
afford  no  direct  evidence  as  to  what  heat  really  is.  This  question 
will  be  considered  later  ;  for  the  present  it  is  sufficient  to  recog- 
nize heat  as  one  of  the  many  forms  in  which  energy  exists. 


Energy  and  Work  1 1 9 

We  have  seen  that  the  kinetic  energy  imparted  to  a  body  by 
an  unbalanced  force  is  the  equivalent  of  the  work  done  by  the 
force.  When  work  is  done  upon  a  body  to  maintain  its  motion 
against  friction,  there  is  no  increase  of  kinetic  energy,  but  heat  is 
developed  ;  and  the  energy  that  appears  in  this  form  is  the  equiv- 
alent of  the  work  done  (see  note  to  Art.  142).  Whenever  one 
surface  moves  upon  another,  heat  is  developed  by  the  friction 
between  them  ;  but  often  so  slowly  that  it  is  dissipated  before 
there  is  appreciable  rise  of  temperature,  and  we  fail  to  notice  it. 
With  rough  surfaces  and  rapid  mo'tion,  however,  the  rise  of  tem- 
perature is  often  considerable.  Many  illustrations  of  this  are 
familiar.  A  match  is  ignited  by  the  heat  generated  in  striking  it. 
A  piece  of  metal  becomes  very  hot  when  ground  on  a  dry  grind- 
stone (Exp.).  When  a  cord  or  a  rope  that  is  grasped  tightly  is 
drawn  rapidly  through  the  hands,  sufficient  heat  is  often  developed 
to  cause  a  burn. 

The  friction  by  which  a  body  is  stopped  changes  or  transforms 
the  kinetic  energy  of  the  body  into  heat  at  the  same  time.  It  is 
for  this  reason  that  the  brakes  of  bicycles,  carriages,  and  cars 
become  hot  when  in  use.  The  same  transformation  of  energy 
occurs  when  the  motion  of  a  body  is  suddenly  stopped  by  impact. 
Thus  a  piece  of  metal  can  be  heated  by  hammering  it,  and  the 
hammer  used  also  becomes  warm  {Exp.),  ^^ullets  are  often  partly 
melted  by  the  heat  developed  when  they  are  stopped  abruptly,  as 
in  striking  a  steel  target  or  a  stone.  • 

149.  Muscular  Enargy.  —  All  the  movements  of  an  animal  are 
due  to  muscular  action,  and  in  this  action  the  muscles  do  work. 
This  work  is  accomplished  by  means  of  the  potential  energy  of  the 
muscles  themselves.  The  amount  of  energy  stored  in  the  muscles 
is  very  great ;  a  horse,  for  example,  can  do  about  two  million  foot- 
pounds of  work  per  hour  for  several  hours.  It  is  evident,  how- 
ever, that  the  amount  is  not  unHmited,  for  any  animal  becomes 
exhausted  after  prolonged  exertion.  The  renewed  supply  of  energy 
comes  from  the  food  eaten. 

Muscular  energy  is  available  for  doing  work  only  through  chem- 


1 20  Energy 

ical  changes  by  which  the  muscles  are  in  part  consumed,  much  as 
fuel  is  consumed  in  a  fire.  Similar  changes  take  place  in  all  the 
organs  of  the  body  while  they  are  performing  their  special  functions, 
and  some  of  the  energy  is  always  liberated  as  heat.  It  is  this  heat 
that  maintains  the  temperature  of  the  body. 

150.  Other  Forms  of  Energy.  —  Electrical  energy,  light,  and 
sound  are  other  forms  of  energy '  which  are  considered  under  the 
corresponding  subdivisions  of  physics.  The  energy  of  a  piece  of 
coal,  as  related  to  the  oxygen  of  the  air  with  which  it  unites  in 
burning,  is  an  example  of  chefiiical  potential  energy.  The  study 
of  this  form  of  energy  belongs  to  chemistry. 

Kinetic  energy  and  energy  of  position  are  classed  together  as 
mechanical  energy ;  and  the  work  done  in  changing  the  kinetic 
energy  of  a  body  or  in  maintaining  its  motion  against  friction  is 
called  mechanical  work. 

151.  The  Transformation  of  Energy.  — A  number  of  instances 
of  the  change  of  energy  from  one  form  into  another  have  already 
been  mentioned  (Arts.  148  and  149).  The  potential  energy  of  a 
drawn  bow  is  transformed  into  kinetic  energy  in  the  act  of  shoot- 
ing an  arrow.  The  potential  energy  of  the  spring  or  the  weight 
of  a  clock  is  very  slowly  transformed  into  heat  by  the  friction  of 
the  moving  parts,  and  a  very  small  portion  into  sound  in  produc- 
ing the  "  tick."  A  body  thrown  upward  loses  kinetic  energy  as  it 
rises,  as  the  result  of  the  negative  work  done  upon  it  by  its  weight, 
and,  at  the  same  time,  gains  an  equal  amount  of  potential  energy 
due  to  its  elevation.  During  the  descent  of  the  body,  its  weight 
does  positive  work  and  the  reverse  transformation  takes  place. 
At  the  end  of  the  fall  the  energy  is  all  kinetic,  and  is  equal  to  the 
kinetic  energy  at  the  start,  except  for  the  resistance  of  the  air, 
which  does  negative  work  upon  the  body  during  both  its  rise  and 
fall.     Muscular  energy  is  transformed  into  kinetic  in  the  act  of 

1  The  words  force  and  energy  were  originally  used  without  a  clear  distinction 
of  meaning,  and  the  different  forms  of  energy  were  commonly  called  "the  forces 
of  nature."  This  usage  still  persists  in  p>opular  language  ;  but  in  science  the  word 
force  is  restricted  to  the  meaning  with  which  the  pupil  is  already  familiar  from 
the  previous  work. 


universitV  1 


Energy  and  Work     \sAt 'rk^^^ 


throwing  a  ball,  into  potential  energy  of  position  in  carrying  a  hod 
of  bricks  up  a  ladder,  and  chiefly  into  heat  in  sandpapering  a  board. 
Other  transformations  of  energy  are  considered  under  the  different 
subdivisions  of  physics. 

152.  The  Dissipation  of  Energy.  —  Since  the  motion  of  all 
bodies  is  opposed  in  a  greater  or  a  less  degree  by  friction  (except 
in  interplanetary  space),  motion  necessarily  involves  the  loss,  or 
dissipation^  of  mechanical  energy  by  its  transformation  into  heat. 
The  energy  thus  transformed  is  said  to  be  dissipated  because  it 
is  no  longer  available  for  doing  useful  work ;  it  is  not  lost  in 
the  sense  that  it  no  longer  exists.  For  example,  if  the  hammer 
of  a  pile  driver  weighs  looo  lb.  and  the  friction  that  must  be 
overcome  in  lifting  it  amounts  to  loo  lb.,  then,  in  raising  the 
hammer  20  ft.,  the  lifting  force  of  iioo  lb.  will  do  20,000  ft.-lb. 
of  work  against  gravity  and  2000  ft.-lb.  of  work  against  friction. 
The  work  done  against  gravity  is  useful,  since,  as  a  result  of  it,  the 
hammer  has  20,000  ft.-lb.  of  available  potential  energy.  The  work 
done  against  friction  is  transformed  into  heat  while  the  hammer  is 
being  raised,  and  cannot  be  further  utilized.  The  work  done  in 
moving  wagons,  street  cars,  and  trains  from  one  place  to  another 
(without  change  of  elevation)  is  done  against  friction,  including 
the  resistance  of  the  air ;  and  the  energy  thus  expended  is  dissi- 
pated as  heat.  When  the  axle  of  a  car  wheel  is  not  properly  oiled 
to  reduce  friction,  the  amount  of  heat  generated  is  so  great  that 
it  causes  a  "  hot  box,"  and  sometimes  even  sets  fire  to  the  car. 
When  bicycles  came  into  general  use,  personal  experience  of  the 
waste  of  energy  due  to  friction  soon  led  to  the  invention  of  ball  bear- 
ings. 

153.  Power. — The  rate  at  which  work  is  done  or  the  rate  at 
which  an  engine  or  other  source  of  energy  is  capable  of  doing 
work  is  called  power.  The  customary  unit  of  power  is  the  horse- 
power (H.  P.),  which  is  equal  to  550  ft.-lb.  or  76  kg.-m.  of  work  per 
second.  Thus  a  twelve-horse-power  engine  working  at  three 
fourths  of  its  full  capacity  is  doing  work  at  the  rate  of  9  H.  P.,  or 
4950  ft.-lb.,  of  work  per  second. 


122  Energy 


PROBLEMS 

1.  Show  from  formula  (20)  that  the  kinetic  energy  of  a  body  is  propor- 
tional (l)  to  its  mass,  (2)  to  the  square  of  its  velocity. 

2.  Two  bodies  have  equal  kinetic  energ)',  but  the  velocity  of  the  second  is 
three  times  that  of  the  first.     How  do  their  masses  compare  ? 

3.  A  body  is  thrown  vertically  upward.  (<?)  What  fraction  of  its  initial 
kinetic  energy  remains  after  it  has  risen  to  one  half  the  height  to  which  it 
will  ascend  ?  (A)  What  fraction  of  its  initial  velocity  remains  ?  (<■)  What 
fraction  of  the  initial  kinetic  energy  and  of  the  initial  velocity  remain  after 
the  body  has  risen  to  three  fourths  of  the  total  height  ? 

4.  (a)  A  boy  starting  at  rest  coasts  on  a  bicycle  down  a  hill  and  up 
another.  If  there  were  no  friction,  how  far  would  he  ascend  the  second  hill 
without  pedaling  ?  (d)  How  would  the  result  be  affected  if  cither  hill  were 
steeper  than  the  other  ? 

5.  An  unbalanced  force  of  50  lb.  acts  through  a  distance  of  4  ft.  upon  a 
mass  of  9  lb.  (a)  What  is  the  work  done  by  the  force  ?  (/')  What  is  the 
kinetic  energy  of  the  body  ?     (<:)  What  is  its  velocity  ? 

6.  A  projectile  weighing  500  lb.  b  fired  from  a  cannon  with  a  velocity  of 
3000  ft.  per  sec.  (a)  What  is  its  kinetic  energy?  (^)  What  was  the  average 
total  pressure  upon  the  projectile  in  firing  it,  if  it  moved  a  distance  of  18  ft. 

before  reaching  the  mouth  of  the  cannon  ? 

y""^         ^\  7.    In  whirling  a  body  round  the  hand  at  the  end  of  a 

string,  the  hand  is  moved  in  a  smaller  circle  in  advance 
of  tht  body  whirled  (Fig.  91).     Show   how   this   imparts 
kinetic  energy  to  the  body. 
V  /  8.    (tf)  Why  is  a  brake  not  heated  if  it  is  applied  with 

such  force  that  the  wheel  slides  along  the  ground  instead  of 
*      *  turning  ?     {b)  Where  will  the  heat  then  be  generated  ? 

9.  (<i)  Is  a  person  doing  work  upon  a  load  that  he  carries  over  level 
ground  ?  (^)  What  is  the  measure  of  the  work  done  upon  the  load  when  it 
is  carried  up  hill  ? 

10.  A  stone  weighing  1.5  lb.  is  thrown  vertically  upward  by  means  of  a 
force  of  20  lb.  acting  through  a  distance  of  3  ft.  How  high  will  it  rise 
above  the  starting  point  ? 

11.  A  body  weighing  15  kg.  falls  vertically  a  distance  of  20  m.  What  is 
its  kinetic  energy  ? 

12.  A  bullet  weighing  5  g.  and  having  a  velocity  of  300  m.  per  sec.  strikes 
a  log  and  penetrates  it  a  distance  of  10  cm.  What  average  resistance  did  the 
bullet  encounter  in  penetrating  the  log? 

13.  (a)   How  much  work  is  done  in  filling  a  reservoir  that  has  a  capacity 


i^) 


The  Simple  Machines  123 

of  1000  cu.  m.  if  the  water  must  be  raised  12  m,  to  discharge  it  into  the  reser- 
voir ?     {b)  How  long  would  it  take  a  twelve-horse-power  engine  to  fill  the 


reservoir 


14.    What  is  the  power  of  an  engine  that  is  capable  of  drawing  a  train  at 
the  rate  of  30  mi.  per  hr.  against  a  resistance  of  6250  lb.  ? 


II.   The  Simple  Machines 

154.  Machines.  —  Any  instrument  or  device  the  purpose  of 
which  is  to  do  work  by  transmitting  energy  from  one  body  to 
another  is  called  a  machine.  In  the  general  sense,  as  here  defined, 
the  term  includes  not  only  the  more  complex  machines,  such  as 
the  steam  engine,  the  printing  press,  and  the  dynamo,  but  also  the 
simplest  tools,  as  the  nutcracker  and  the  pocketknife.  In  many 
machines  the  energy  is  transformed  as  well  as  transferred.  The 
dynamo,  for  example,  transforms  mechanical  into  electrical  energy. 
In  the  steam  engine  the  chemical  potential  energy  of  the  fuel 
undergoes  several  transformations,  and,  in  the  end,  is  transferred 
as  mechanical  energy  to  the  driving  wheels. 

All  machines,  however  complicated,  are  but  modifications  and 
combinations  of  one  or  more  of  the  six  simple ' machines  or 
mechanical  powers.  These  are  the  lever,  the  wheel  and  axle,  the 
pulley,  the  inclined  plane,  the  wedge,  and  the  screw.  The  study 
of  the  simple  machines  will  afford  a  review  and  application  of 
much  that  has  already  been  learned  concerning  force  and  work. 

155.  The  Lever.  —  A  lever  is  essentially  a  bar  or  rod  capable  of 
turning  about  some  fixed  support  or  axis,  called  the  fulcrum.  A 
crowbar  is  used  as  a  lever  in  moving  heavy  objects  (Fig.  55).  In 
using  a  lever,  a  force,  called  the  effort^  applied  at  some  point  of 
it,  causes  it  to  exert  a  force  at  some  other  point  upon  the  object 
to  be  moved.  The  force  exerted  upon  the  lever  by  this  object 
is  called  the  resistance.  The  resistance  is  generally,  though  not 
always,  due  to  the  weight  of  the  body  upon  which  the  lever  acts. 

1  The  effort  is  commonly  called  the  power  ;  but  this  use  of  the  word  is  objec- 
tionable since  power  also  means  the  rate  at  which  work  is  done. 


1 24  Energy 

There  are  three  classes  of  levers.  In  levers  of  the  first  class 
(Fig.  92)  the  fulcrum  O  is  between  the  points  of  application  of 
the  effort  /  and  the  resistance  Fy  in  levers  of  the  second  class 

^-^-^ n «.  ?— ■ ^- V 

\r      Fig.  9a. 


F  Fig.  93.  Fig.  94. 


I 


(Fig.  93)  the  resistance  acts  between  the  fulcrum  and  the  effort ; 
in  levers  of  the  third  class  (Fig.  94)  the  effort  is  applied  between 
the  fulcnim  and  the  resistance.  The  three  classes  are  illustrated 
in  laboratory  Exercise  17.  (Refer  to  the  exercise  and  identify 
the  illustrations.) 

In  all  cases  the  effort  and  the  resistance  tend  to  turn  the  lever 
in  opposite  directions  about  the  fulcrum ;  hence,  when  the  lever 
is  in  equilibrium,  the  moments  of  the  effort  and  the  resistance  are 
equal  (Art.  71).  If  a  and /^  denote  respectively  the  perpendic- 
ular distances  from  the  fulcrum  to  the  lines  of  action  of/  and  Fj 
the  condition  necessary  for  equilibrium  is 

fa  =  FA^oi  F'.f '.\a\A.  (21) 

When  the  resistance  is  a  weight,  it  is  commonly  denoted  by  Wy 
in  which  case  the  condition  of  ecjuilibrium  is  written/z  =  WA, 

When  a  lever  is  in  action,  even  with  uniform  motion,  the 
moment  of  the  effort  is  very  slightly  greater  than  the  moment  of 
the  resistance,  on  account  of  friction ;  but  the  small  difference  is 
commonly  disregarded,  and  in  solving  problems,  the  condition  of 
equilibrium  is  assumed  to  hold  when  the  lever  is  in  action.  The 
same  assumption  is  made  in  studying  the  other  simple  machines. 

166.  Applications  of  the  Lever.  —  The  lever  in  different  forms 
is  adapted  to  various  special  uses.  In  most  cases  it  enables  a 
given  force  to  overcome  a  resistance  several  times  greater  than 
itself;  for  the  effort  and  the  resistance  are  inversely  proportional 
to  their  arms,  and  the  arms  may  be  taken  in  any  desired  ratio. 
Thus  a  stone  that  requires  a  force  of  600  lb.  to  move  it  can  be 


The  Simple  Machines  125 

moved  with  a  crowbar  by  exerting  upon  the  latter  a  force  of  100 

lb.,  when  the  fulcrum  is  placed  so  that  the 

arm  of  the  effort  is  six  times  as  long  as  the 

arm   of  the  resistance.     The   advantage 

thus  derived  is  referred  to  as  a  gain  of  Fig.  95. 

force.    The  same  advantage  is  afforded  by  forceps,  pincers,  wire 

^ cutters  (Fig.  95),  and  nutcrackers  (Fig. 

^^ ^"^ll^m    96)  ;    all   of   which   are    double   levers 

\H^^=^^^''^^^^^    having  the  arm  of  the  effort  longer  than 

_  „  _^  the  arm  of  the  resistance. 

rIG.  90. 

Other  forms  of  the  lever  are  designed 
with  reference  solely  to  convenience  or  adaptability  to  the  work 
to  be  done,  the  relative  value  of  effort  and  resistance  being  un- 
important. Tweezers,  coal  tongs,  sugar  tongs,  and  scissors  are 
examples.  With  all  except  the  last  the  effort  is  necessarily  greater 
than  the  resistance.     (Why?) 

In  certain  applications  of  the  lever  the  end  sought  is  a  gain  of 
speed  and  distance  ;  that  is,  the  conditions  are  such  that  the  point 
of  application  of  the  resistance  moves  faster  and  farther  than  the 
point  of  application  of  the  effort.  This  is  well  illustrated  by  the 
movements  of  our  bodies  and  the  bodies  of  animals  in  general. 
The  movable  parts  of  the  skeleton  are  levers ;  the  joints  are  the 
fulcrums.  The  muscles  are  attached 
to  the  bones  near  the  joints  by  means 
of  tendons.  A  muscle  acts  by  con- 
tracting or  shortening.  This  causes 
the  bone  to  which  it  is  attached  to 
move,  and  the  farther  extremity  of  _  ^ 

the   bone   moves   much   faster  and 

farther  than  the  point  to  which  the  muscle  is  attached.  Figure  97 
shows  the  action  of  the  forearm  in  lifting  a  weight  in  the  hand. 
It  is  evident  that  a  slight  shortening  of  the  muscle  is  sufficient  to 
raise  the  hand  a  foot  or  more.  There  is,  however,  a  corre- 
sponding loss  of  force  ;  that  is,  the  effort  is  much  greater  than  the 
resistance.     (Why?) 


1 26  Energy 

A  curved  or  a  bent  lever  is  often  more  convenient  than  a  straight 
one.  A  claw  hammer  used  in  drawing  a  nail  is  an  example  of  a 
bent  lever  (Fig.  56).  The  condition  of  equilibrium  is  the  same 
whatever  the  shape  of  the  lever,  it  being  understood  that  the  arms 
are  always  the  perpendicular  distances  from  the  fulcrum  to  the 
lines  of  action  of  the  effort  and  the  resistance. 

157.  Mechanical  Advantage.  —  The  ratio  of  the  resistance  to 
the  effort  {F.f)  is  called  the  mechanical  adi^antage.  The  princi- 
pal problem  in  the  study  of  a  simple  machine  is  to  find  the  value 
of  this  ratio  in  terms  of  certain  dimensions  of  the  machine.  It 
follows  from  the  condition  of  equihbrium  that  the  mechanical  ad- 
vantage of  a  ln>er  is  the  ratio  0/  the  arm  0/  the  effort  to  the  arm  of 
the  resistance  {a:  A).     This  ratio  is  frequently  called  the  leverage, 

158.  Work  done  with  a  Lever.  —  Suppose  a  constant  effort/, 
acting  at  the  end  of  a  lever  of  the  first  class  (Fig.  98),  to  move 

the  lever  through  a  certain  angle  against 
^'    '^^  ^  "  a  constant  resistance  7%  applied  at   the 


_ — ,_ _  _  ,  „,.^ — 

"^-^^  ^i^^  other  end.  Suppose  further  that/" and  ^ 
""-''  act  at  right  angles  to  the  lever  through- 
out the  motion.  I^t //denote  the  distance 
through  which  the  effort  acts  and  Z>  the  distance  through  which 
the  resistance  is  overcome.  Then  fl  is  the  work  done  by  the 
effort  and  FD  the  work  done  against  the  resistance ;  or,  we  may 
say,  fl  measures  the  energy  transferred  to  the  lever  by  the  agent 
producing  the  motion,  and  FZ>  measures  the  energy  transferred 
by  the  lever  to  the  object  moved.     Formula  (21)  is 

F:f::a:A 
and,  by  geometry,  d:  D::a:  A; 

hence,  F:f::d:Df 

or,  fd=^FD. 

This  means  that,  neglecting  friction,  the  lever  transmits  energy 
from  the  agent  to  the  body  acted  upon  without  gain  or  loss. 
This  is  easily  proved  to  be  true  for  levers  of  the  second  and  third 
classes  also;  and  it  can  be  shown  to  hold  in  all  cases,  however  the 


The  Simple  Machines 


127 


forces  may  vary  in  magnitude  or  direction.  There  may  be  a  gain 
of  force  by  the  use  of  a  lever ^  but  never  a  gain  of  etiergy.  There  is 
always,  in  fact,  a  slight  loss  or  waste  of  energy  due  to  friction ; 
that  is,// is  slightly  greater  than /^Z?. 


PROBLEMS 

1.  In  using  scissors  is  greiater  force  required  when  the  cutting  is  done  near 
the  tips  df  the  blades  or  near  the  handles  ?     Why  ? 

2.  Classify  the  following  levers,  and  state  in  each  case  whether  the  effort 
is  greater  or  less  than  the  resistance  :  the  wheelbarrow,  oar,  fishing  rod, 
equal-arm  balance,  steelyard,  nutcracker. 

3.  Use  a  pencil  as  a  lever  of  the  first  class  to  move  a  book ;  also  as  a  lever 
of  the  second  class. 

4.  In  which  class  or  classes  of  levers  is  the  effort  necessarily  less  than  the 
resistance  ?  greater  than  the  resistance  ?  In  which  may  it  be  either  greater 
or  less  ? 

5.  Prove  that  fit=  ^D  when  a  weight   W  is  raised  through  a  vertical 
distance  D  by  means  of  a  force  /  acting  ver- 
tically through  a  distance  </  (Fig.  99). 

6.  (rt)  If  a  stone  offers  a  resistance  of  850  lb., 
what  leverage  will  be  required  to  move  it  by 
means  of  a  force  of  125  lb.?  (Ji)  If  the  stone  is 
moved  by  a  crowbar  5  ft.  long  used  as  a  lever 
of  the  first  class,  the  effort  and  the  resistance 
being  applied  at  the  ends,  where  is  the  fulcrum? 

7.  Show  that  when  there  is  a  gain  of  force  by  the  use  of  a  lever  there  is  a 
proportional  loss  of  speed  and  distance  ;  and  that  when  there  is  a  gain  of 
distance  there  is  a  proportional  loss  of  force. 

159.   The  Wheel  and  Axle.  —  The  wheel  and  axle  (Figs.  100 

and  loi)  consists  of  a 
wheel  and  cylinder  fas- 
tened together  and  ca- 
pable of  turning  on  the 
same  axis.  The  effort  is 
applied  at  the  circumfer- 
ence of  the  wheel,  and 
the  resistance  at  the  cir- 
FiG.  100.  cumference  of  the  axle. 


Fig.  99 


128 


Energy 


The  machine  may  be  regarded  as  a  continuously  acting  lever  of 
the  first  or  the  second  class,  depending  upon  whether  the  effort 
and  the  resistance  are  applied  on  opposite  sides  or  the  same  side 
of  the  axis.    The  condition  necessary  for  equilibrium  is 

/r=  /ra,  or  IV: /: :  r :  ^,  (22) 

in  which  r  and  /^  denote  the  radius  of  the  wheel  and  the  axle 
respectively. 

TAf  mechanical  advantage  of  the  wheel  and  axle  Is,  therefore, 
the  ratio  of  the  radius  of  the  wheel  to  the  radius  of  the  axle  (r  .•  R) . 

Laboratory  Exercise  2j. 

160.  Work  done  with  a  Wheel  and  Axle.  —  During  one  com- 
plete turn  of  the  wheel  the  effort  acts  once  round  its  circum- 
ference, Cy  and  the  weight  is  raised  a  distance  equal  to  the 
circumference  of  the  axle,  C.  Whatever  the  angle  may  be  through 
which  the  wheel  is  turned,  the  distance  d  through  which  the  effort 
acts  and  the  distance  D  through  which  the  weight  is  raised  are  in 
the  same  ratio  as  for  one  revolution  ;  that  is, 

d'.D=^c.C=r'.R. 
But  W:f:.r:R; 

hence,  W:f'..d:D, 

or,  //=  WD. 

That  is,  the  wheel  and  axle,  like  the  lever,  transmits  energy 

without  loss  or  gain,  except  in  so 
far  as  there  is  loss  due  to  friction. 
161.  The  Windlass  (Fig.  102) 
is  a  modified  form  of  the  wheel  and 
axle,  and  acts  upon  the  same  prin- 
ciple. As  the  crank  is  turned,  the 
handle  describes  a  circumference ; 
hence  the  crank  is  equivalent  to  a 
wheel  having  a  radius  equal  to  the 
arm  of  the  crank. 
102.  162.   The  Fixed  Pulley.  — A  pul- 


The  Simple  Machines 


129 


ley  that  turns  on  a  stationary  axis  is  called  a  fixed  pulley, 
be  regarded  as  an  endless  or  continuous  lever 
of  the  first  class  having  equal  arms.  Since  the 
arms  are  equal,  the  effort  and  resistance  are 
also  equal  (/=  IV).  The  same  conclusion 
follows  from  the  fact  that,  except  for  a  slight 
possible  difference  due  to  friction,  the  parts  of 
the  cord  on  the  two  sides  of  the  pulley  are 
necessarily  under  the  same  tension. 


It  may 


E 

^o^ 


Fig.  103. 


Fig.  104. 


The  only  advantage  gained  by  the  use  of 
one  or  more  fixed  pulleys  is  a  change  in  the 
direction  of  the  effort  (Fig.  103). 

163.  The  Movable  Pulley.  —  Figure  104  represents  the  com- 
bination of  a  fixed  and  a  movable  pulley.  The  fixed  pulley  serves 
the  same  purpose  as  when  acting  alone.  The  weight,  which  is 
applied  at  the  axis  of  the  movable  pulley,  is  balanced  by  the  two 
equal  upward  pulls  of  the  two  parts  of  the  cord  that  support  the 
pulley.  These  pulls  are  equal  to  the  effort  /,  and  their  sum  is 
equal  to  the  weight  (including  the  weight  of  the  supported  pulley)  ; 

that  is, 

2/=  rr,  and  IV  :/=2. 

The  mechanical  advantage  of  the  movable  pulley  is  therefore 
two. 

164.  Systems  of  Pulleys.  —  A  combination  of  several  fixed 
and  movable  pulleys  is  frequently  used  where  great  resistances  are 


130  Energy 

to  be  overcome.  A  variety  of  arrangements 
is  possible ;  but  we  shall  consider  only  the  one 
generally  used,  namely,  that  in  which  one  con- 
tinuous cord  or  rope  passes  alternately  round 
the  pulleys  in  a  fixed  and  a  movable  block,  as 
shown  in  the  figures. 

In  finding  the   general  condition  of  equi- 
librium it  is  assumed  that  all  parts  of  the  cord 
are  under  ecjual  tension.    The  weight  is  there- 
fore supported  by  as  many  equal  parallel  forces 
as  there  are  parts  of  the  cord  supporting  the 
Fic.  105.  movable  block.     Let  this  number  be  denoted 
by  «  («  is  3  in  Fig.  105  and  6  in  Fig.  106)  ; 
then,  since  the  tension  is  equal  to  the  effort,  the  con- 
dition of  equilibrium  is  expressed  by  the  equation 

nf  =  W,  or  W  \f=  n.  (23)        I  lll'r 

Tht  mechanical  advantage  of  such  a  system  of  pulleys  '  '    ' 

is  therefore  equal  to  the  number  of  parts  of  the  cord  supporting  the 
movable  block. 

Laboratory  Exercise  24. 

165.  Work  done  with  Pulleys.  —  The  ratio  of  the  distance  d 
through  which  the  effort  acts  to  the  distance  D  through  which  the 
weight  is  raised  is  easily  determined  for  any  combination  of  pul- 
leys, from  the  fact  that  the  length  of  the  whole  cord  remains  con- 
stant. With  a  single  fixed  pulley  the  part  of  the  cord  to  which 
the  effort  is  a'pplied  lengthens  as  much  as  the  part  to  which  the 
weight  is  attached  shortens  {d  =  D) -,  hence,  since/ = /%  we 
have//=  WD, 

With  any  number  of  fixed  and  movable  pulleys  connected  by 
one  continuous  cord,  when  the  weight  is  raised  a  distance  D, 
each  of  the  n  parts  of  the  cord  supporting  the  movable  block  is 
shortened  by  an  equal  amount,  and  the  length  of  the  part  to  which 
the  effort  is  applied  is  increased  by  nD  {i.e.  d  =  nD)  ;  hence, 
since /=  IF-i-  n  (equation  23),  it  follows  that/^/=  WD, 


The  Simple  Machines  131 

That  is,  neglecting  friction,  any  combination  of  pulleys  transmits 
energy  without  gain  or  loss  from  the  agent  to  the  body  acted 
upon. 

PROBLEMS 

1.  (a)  The  radius  of  a  wheel  is  40  cm.  and  the  radius  of  the  axle  12  cm. 
Neglecting  friction,  what  effort  is  required  to  raise  a  load  of  150  kg.? 
{d)  Through  what  distance  does  the  effort  act  in  raising  the  load  35  m.? 
(c)  How  much  work  b  done  by  the  effort  ?  (^)  How  much  is  done  against 
the  weight  of  the  load  ? 

2.  Establish  the  condition  necessary  for  the  equilibrium  of  a  single  mov- 
able pulley,  regarding  it  as  a  lever  of  the  second  class.  Draw  a  figure  to 
illustrate. 

3.  Draw  a  figure  of  a  system  of  pulleys  such  that  the  mechanical  advantage 
is  seven  ;  such  that  it  is  eight. 

166.   The  Inclined  Plane.  —  The  principal  use  of  the  inclined 

plane  is  to  raise  heavy  bodies 

that   can  be   rolled.     When 

a  barrel  of  flour  is  placed  in 

a  wagon  by  rolling  it  up  a 

heavy    plank,    the   plank    is 

used  as   an   inclined   plane. 

The  effort  in   such  cases  is 

applied  parallel  to  the  ])lane, 

and  is  sufficient  to  maintain 

equilibrium  if  it  is  equal  to  the  component  of  the  weight  of  the 

body  acting  parallel  to  the  plane  in  the  opposite  direction  (Fig. 

107).     It  follows  from  this  (see  Art.  143)  that  the  condition  of 

equilibrium  is 

lV:/::L:/f,  (24) 

in  which  Z  denotes  the  length  of  the  plane,  AB,  and  H  its  height, 
BC, 

Hence,  when  the  effort  is  applied  parallel  to  an  inclined  plane  ^ 
the  mechanical  advantage  is  the  ratio  of  the  length  of  the  plane  to 
its  height. 

Laboratory  Exercise  25. 


Fic.  107, 


132  Energy 

167.  Work  done  with  an  Inclined  Plane.  —  Let  D  (Fig.  107) 
denote  the  vertical  distance  through  which  a  body  is  raised  by 
rolling  it  a  distance,  //,  up  an  inclined  plane  {d  being  any 
fraction  of  the  length  of  the  plane)  ;  then  the  work  done  by 
the  effort  is  fd  and  the  work  done  upon  the  body  (against 
gravity)  is  WD. 

From  similar  triangles         d\  D\\  L\  H\ 
hence,  combining  with  (24),   W\f\  \  d.  D^ 
or,  //=  WD, 

That  is,  disregarding  friction,  the  same  amount  of  work  is  done 
whether  a  body  be  rolled  up  an  inclined  plane  or  lifted  verti- 
cally to  the  same  height. 

168.  No  Machine  creates  Energy.  —  It  has  been  shown  that, 
disregarding  friction,  the  work  done  by  any  machine  that  we  have 
studied  is  exactly  equal  to  the  work  done  upon  the  machine.  The 
significant  fact  to  be  noted  is  that  there  is  in  no  case  a  gain  of 
work,  or  a  creation  of  energy.  This  is  true  of  all  machines  without 
exception.  No  machine  has  ever  been  invented  that  was  an 
original  source  of  even  the  smallest  amount  of  energy ;  on  the 
contrary,  there  are  the  best  of  reasons  for  believing  that  no  such 
machine  is  possible  (Art.  271). 

169.  Efficiency  of  Machines.  —  No  machine,  however  perfect, 
does  useful  work  that  is  the  full  equivalent  of  the  work  done  upon 
it.  The  principal  cause  of  waste  is  friction,  by  which  a  portion  of 
the  energy  transferred  to  the  machine  is  transformed  into  useless 
heat  (Art.  152).  Further  waste  of  energy  results  from  the  bend- 
ing and  vibration  of  the  parts  of  a  machine  and  from  the  stiffness 
of  ropes  and  belts. 

The  ratio  of  the  useful  work  done  by  a  machine  to  the  total 
energy  transferred  to  it  is  called  the  efficiency  of  the  machine. 
The  efficiency  of  an  ideally  perfect  machine  would  be  unity  or 
100  per  cent.  A  delicately  balanced  lever,  such  as  the  beam  of 
a  sensitive  balance,  is  the  nearest  approach  to  such  a  machine. 
In  general,  the  greater  the  complexity  of  a  machine,  the  less  its 


The  Simple  Machines 


133 


Fig.  108. 


efficiency.  The  pupil  doubtless  observed  in  experimenting  with 
pulleys  that  friction  becomes  greater  as  the  number  of  pulleys  is 
increased.  With  five  or  six  pulleys  in  the  combination,  the  effi- 
ciency will  hardly  exceed  75  per  cent  and  may  be  considerably 
less. 

170.  The  Wedge.  — The  wedge  may  be  regarded  as  a  movable 
inclined  plane.  It  is  used  for  separating  surfaces  asjainst  great 
resistance,  as  in  splitting  logs  and  timbers. 
The  motion  of  a  wedge  is  opposed  by  very 
great  sliding  friction,  and  its  efficiency  is 
correspondingly  low;  yet  this  friction  is 
useful  and  indeed  necessary,  for  it  keeps 
the  wedge  from  slipping  out  of  place  dur- 
ing the  intervals  between  the  blows  that 
drive  it  farther  in.  The  wedge  utilizes 
a  great  force  acting  for  a  very  short  time, 
and  is  thus  enabled  to  overcome  a  great 
resistance  through  a  small  distance.  Hence,  though  slow  in 
action,  it  is  a  very  powerful  machine  (Fig.  108). 

No  definite  law  can  be  given  that  holds  even  approxi- 
mately in  the  use  of  the  wedge.  The  thinner  the  wedge, 
however,  the  greater  is  its  mechanical  advantage.  The  ax, 
the  knife,  and  the  chisel  are  forms  of  the  wedge  adapted  to 
special  uses. 

171.  The  Screw. —The  screw  is  a  modification  of  the 
inclined  plane,  as  will  be  seen  by  winding  a  paper  triangle 
about  a  pencil  (Fig.  109).  The  ridge  running  spirally  round 
a  screw  is  called  the  thread.  The  distance  between  suc- 
cessive turns  of  the  thread,  measured  parallel  to  the  axis  of 
the  screw,  is  called  the  pitch  of  the  screw.  The  screw 
turns  in  a  block,  called  the  nut,  which  is  provided  with  a 
spiral  groove  to  receive  the  thread. 

When  the  screw  is  used  as  a  part  of  a  machine, 

the  effort  is  generally  applied  to  it  by  means  di 

Fig.  109.       a  lever  or  a  wheel.     The  lifting  jack  (Fig.  no) 


1 34  Energy 

is  an  example  of  the  first  case,  the  copying  press  (Fig.  iii)  of  the 
second.     During  one  complete  turn  of  the  lever  or  the  wheel  the 


Fiii.  na  Fig.  hi. 

screw  advances  through  a  distance  equal  to  the  pitch  of  the 
screw,  p.  The  work  done  by  the  effort  during  one  turn  is  2  trrf,  r 
being  the  radius  of  the  wheel  or  the  arm  of  the  lever ;  and  the 
work  done  against  the  resistance,  Fy  is  Fp.  Hence,  if  friction 
were  negligible,  we  should  have 

2irr/=^Fp.  (25) 

This  is  sometimes  given  as  the  law  of  the  screw,  from  which  the 
mechanical  advantage  is  determined ;  but  the  error  is  very  large. 
In  fact,  after  the  screw  has  been  turned  into  any  position,  friction 
alone  is  generally  much  more  than  sufficient  to  hold  it  in  place 
against  the  resistance.  Friction  is  for  this  reason  useful  and  nec- 
essary in  most  applications  of  the  screw,  as  it  is  for  the  wedge. 
Without  it,  woodwork  and  the  parts  of  machines  could  not  be 
held  together  by  means  of  screws. 

172.  The  Bicycle. — The  bicycle  is  a  familiar  example  of  a 
compound  machine  in  which  a  number  of  the  foregoing  principles 
are  applied.  The  cranks  and  the  large  sprocket  wheel  are  a  modi- 
fied wheel  and  axle.  The  motion  of  the  larger  sprocket  wheel  is 
communicated  by  means  of  the  chain  to  a  smaller  one  attached 


The  Simple  Machines  135 

to  the  rear  wheel  of  the  machine  and  turning  upon  the  same 
axle.  Since  the  chain  in  turning  passes  by  the  same  number  of 
sprockets  on  both  wheels,  the  smaller  one  turns  as  many  times 
faster  than  the  larger  as  the  number  of  sprockets  on  the  smaller  is 
contained  in  the  number  on  the  larger.  If  this  ratio  is  three,  for 
example,  the  smaller  sprocket  wheel  makes  three  complete  turns 
while  the  larger  makes  one.  But  the  larger  wheel  turns  at  the 
same  rate  as  the  cranks,  and  the  smaller  at  the  same  rate  as 
the  rear  wheel  of  the  machine.  Hence,  in  the  case  supposed,  the 
rear  wheel  makes  three  revolutions  while  the  cranks  make  one. 
Thus  the  bicycle  travels  as  far  during  one  revolution  of  the  cranks 
as  it  would  if  they  were  attached  directly  (as  in  the  old  style 
"  ordinary  ")  to  a  wheel  having  a  diameter  three  times  as  great. 
The  diameter  of  this  equivalent  wheel,  measured  in  inches,  is 
called  the^^^r  of  the  bicycle.  Thus,  if  the  ratio  of  the  sprockets 
is  three,  the  gear  is  3  X  28,  or  84,  the  diameter  of  the  wheels  of  a 
"safety  "  being  28  in.  With  this  gear  and  cranks  having  a  radius 
of  7  in.,  the  speed  of  the  bicycle  is  84  -h  14,  or  6  times  as  great 
as  the  speed  of  the  cranks  round  the  axle. 

PROBLEMS 

1.  A  block  at  rest  upon  a  board  i  m.  long  begins  to  slide  when  the  board 
is  inclined  so  that  its  higher  end  is  40  cm.  above  the  other.  The  friction  is 
what  per  cent  of  the  weight  of  the  block  ? 

2.  Find  by  geometry  the  mechanical  advantage  of  the  inclined  plane  when 
the  effort  is  applied  parallel  to  the  base  of  the  plane,  i.e.  horizontally. 

3.  Assuming  a  loss  of  20%  (efficiency  80%),  what  force  is  required  to  haul 
a  load  of  2  tons  (including  the  weight  of  the  wagon)  up  a  grade  such  that  the 
ascent  is  one  foot  in  a  distance  of  10  ft.  ? 

4.  What  effort  will  be  required  to  raise  a  weight  of  200  kg.  with  a  wheel 
and  axle  the  efficiency  of  which  is  90  % ;  the  radius  of  the  wheel  being  42  cm. 
and  the  radius  of  the  axle  14  cm.  ? 

5.  The  weight  in  the  preceding  problem  is  raised  25  m.  Find  :  {a)  the 
work  done  upon  the  machine  ;  {b)  the  work  done  against  gravity ;  {c)  the 
energy  wasted. 

6.  What  do  you  gather  from  Arts.  168  and  169  concerning  the  possibility 

of  a  **  perpetual  motion  "  machine  ? 


136  Energy 

7.  Is  a  sewing  machine  constructed  for  a  gain  of  speed  or  of  force  ?  By 
what  aicchanism  is  this  gain  accomplished  ? 

8.  How  long  must  a  plank  be  in  order  that  it  may  be  used  as  an  inclined 
plane  to  raise  a  barrel  weighing  150  lb.  to  an  elevation  of  4  ft.  by  a  force  of 
40  lb.,  assuming  an  efficiency  of  ic»  %  ? 

9.  A  lifting  jack,  the  screw  of  which  has  a  pitch  of  }  in.,  is  used  to  raise  a 
weight  of  10  tons.  The  effort  is  applied  at  the  end  of  a  lever  2  ft.  long,  and 
the  efficiency  is  40  %.     Kind  the  efiort. 


CHAPTER  VII 

SOME  PROPERTIES  OF  MATTER 

I.  The  Structure  of    Matter 

173.  Properties  of  Matter. — The  properties  of  matter  are  of 
two  classes — general  or  universal  and  specific  or  characteristic. 
General  properties  are  common  to  all  matter.  Among  the  most 
important  are  extensioUy  inertia,  divisibility,  porosity,  compressibility ^ 
elasticity,  indestructibility.  Specific  properties  are  those  possessed 
by  certain  kinds  or  slates  of  matter,  but  not  by  all;  such  are 
rigidity,  fluidity,  brittleness,  hardness,  tenacity,  transparency,  color, 
odor,  etc.  It  is  only  by  means  of  differences  in  the  properties  of 
substances  that  we  can  distinguish  them  from  one  another. 

The  laws  of  liquid  pressure  are  direct  consequences  of  the 
characteristic  properties  of  liquids,  namely,  their  fluidity  and  their 
very  nearly  perfect  incompressibility.  Similarly,  the  laws  of  gases 
follow  from  the  fluidity  and  the  unlimited  expansibility  of  gases. 
In  dealing  with  solids,  we  have  assumed  that  the  behavior  of  a 
body  under  the  action  of  forces  is  not  affected  by  bending  or  other 
distortion  of  the  body,  however  the  forces  may  be  applied ;  i.e. 
solids  were  assumed  to  be  perfectly  rigid.  Thus  the  mechanics 
of  solids,  liquids,  and  gases  is  largely  dependent  upon  the  dis- 
tinguishing properties  of  the  three  states  of  matter. 

Extension  and  inertia  have  been  sufficiently  considered  in  pre- 
vious chapters.  A  study  of  the  other  general  properties  mentioned 
above  suggests  important  inferences  concerning  the  structure  of 
matter. 

174.  Divisibility.  —  Any  substance  can  be  divided  into  parts 
or  fragments ;  and  the  division  can  be  carried  so  far  that  the 
individual  particles  are  invisible  to  the  unaided  eye  and  are  barely 

137 


138  Some  Properties  of  Matter 

visible  with  the  aid  of  a  microscope.  Suppose  a  grain  of  salt  to 
be  pulverized  till  a  microscopic  particle  is  obtained.  Subdivision 
can  be  carried  no  farther  by  mechanical  means ;  but  the  particle 
can  be  dissolved  in  a  drop  of  water,  and,  although  what  happens 
to  it  in  this  process  is  entirely  invisible,  many  facts  that  cannot  be 
presented  here  point  to  the  conclusion  that,  in  dissolving,  //  is  scp- 
arattd  into  millions  of  parts. 

The  almost  inconceivable  divisibility  of  matter  is  well  illustrated 
by  dissolving  a  minute  grain  of  potassium  permanganate  or  of  aniline 
dye  in  several  quarts  of  water.  The  whole  will  be  appreciably 
colored,  showing  the  presence  of  portions  of  the  substance  in  every 
drop  {Ex'p.).  Numerous  illustrations  might  be  given  from  daily 
life.  After  some  years  of  service  a  gold  or  a  silver  coin  is  found 
to  be  considerably  worn;  yet  no  visible  portion  is  lost  at  any 
time.  Similarly,  a  knife  becomes  dull  from  the  gradual  loss  of 
invisible  particles  at  and  near  its  edge.  The  tenth  part  of  a  grain 
of  musk  will  continue  for  years  to  fill  a  room  with  its  odoriferous 
particles,  and  at  the  end  of  that  time  will  scarcely  be  diminished 
in  weight. 

175.  Molecules.  —  Since  the  divisibility  of  matter  far  exceeds 
our  power  of  vision,  it  is  only  by  reasoning  and  inference  based 
upon  other  facts  that  it  can  be  determined  whether  there  is  any 
limit  to  divisibility.  Many  facts  in  chemistry  indicate  that  there 
is  such  a  limit.  In  other  words,  there  is  such  a  thing  as  the 
smallest  possible  particle  of  any  substance.  This  smallest  particle 
is  called  a  molecule.  All  molecules  of  the  same  substance  are 
exactly  equal  in  size  and  weight,  and  are  alike  in  every  respect. 

It  is  estimated  that,  in  a  cubic  millimeter  of  any  gas  at  atmos- 
pheric pressure  and  at  ordinary  temperatures,  there  are  some- 
where about  20,000,000,000,000,000  molecules ;  and  that,  in  an 
equal  volume  of  a  liquid  or  a  solid,  the  number  is  hundreds  or 
thousands  of  times  as  great. 

176.  Porosity.  —  Openings  within  a  solid  are  called  sensible 
pores  whether  large  enough  to  be  visible  with  the  unaided  eye,  as 
the  holes  in  a  sponge  or  a  piece  of  bread,  or  so  small  as  to  be 


The  Structure  of  Matter 


139 


visible  only  with  a  microscope,  as  the  pores  of  blotting  paper,  brick, 
and  unglazed  earthenware.  The  spaces  or  openings  between  the 
molecules  of  a  body  are  cdiWtdi  p/iysical pores.  The  most  powerful 
microscope  is  incapable  of  rendering  them  visible ;  but  their 
existence  has  been  repeatedly  demonstrated  by  experiment. 

In  1 66 1  some  academicians  at  Florence  subjected  to  great 
pressure  a  thin  globe  of  gold  filled  with  water.  Their  object  was 
to  determine  whether  water  was  compressible ;  they  discovered 
the  porosity  of  gold  instead,  for  "  the  water  forced  its  way  through 
the  pores  of  the  gold,  and  stood  on  the  outside  of  the  globe  like 
dew."  The  experiment  has  been  repeated  with  globes  of  other 
metals,  and  always  with  the  same  result  —  the  metals  were  proved 
to  be  porous.  A  pressure  of  less  than  one  atmosphere  is  sufficient 
to  force  mercury  through  a  thick  piece  of  leather  {Exp.) ;  and, 
with  a  pressure  of  4000  atmospheres,  mercury  has  been  forced 
through  three  inches  of  solid  steel. 

With  the  exception  of  glass  and  other  vitreous  bodies,  similar 
experiments  have  not  failed  to  prove  the  porosity  of  any  solid. 

177.  Porosity  a  General  Property  of  Matter. — The  fact  that 
the  volume  of  any  substance,  whether  solid,  liquid,  or  gaseous,  can 
be  changed  either  by  change  of  temperature  or  change  of  pressure 
is  both  a  proof  of  the  porosity  of  all  matter  and  a  consequence 
of  this  property.  Such  changes  of  volume  are  considered  in  the 
following  paragraphs.  Further  striking  evidence  of  the  porosity 
of  certain  liquids  is  afforded  by  the  change  of  volume  that  occurs 
in  mixing  them.  Thus,  when  equal  or  nearly  equal  volumes  of 
water  and  strong  alcohol  are  mixed,  it  is  found  that  the  volume 
of  the  mixture  is  considerably  less  than  the  sum  of  the  volumes  of 
the  liquids  before  they  are  mixed,^  proving  that  the  molecules 
of  water  and  alcohol  fit  together  more  closely  than  the  molecules 
of  one  or  the  other,  or  possibly  both,  of  the  liquids  do  among 

1  This  is  readily  shown  by  filling  a  long,  slender  test  tube  half  full  of  water  and 
adding  alcohol  carefully,  to  avoid  mixing,  till  the  tube  is  nearly  full.  Mark  the 
height  of  the  alcohol  with  a  rubber  band,  close  the  mouth  pf  the  tube  with  the  thumb, 
and  shake. 


140  Some  Properties  of  Matter 

themselves  {Exp.).  The  idea  is  iUustrated  on  an  enormously 
magnified  scale  by  mixing  together  a  quantity  of  sand  and  an  equal 
volume  of  coarse  shot.  A  loss  of  volume  occurs,  due  to  the  filling 
of  the  spaces  between  the  shot  by  the  smaller  grains  of  sand.  The 
loss  of  volume  on  mixing  the  alcohol  and  water  is  regarded  as 
conclusive  evidence  that  there  are  spaces  void  of  matter  between 
the  molecules  of  the  liquids. 

A  similar  shrinkage  occurs  in  mixing  strong  sulphuric  acid  and 
water,  and  also  when  certain  solids,  as  sugar  or  salt,  are  dissolved 
in  water. 

178.  Compressibility.  —  All  matter  is  more  or  less  compressible. 
The  compressibility  of  gases,  as  expressed  by  Boyle's  law,  has 
already  been  considered.  The  compressibility  of  liquids  and 
solids,  although  extremely  small,  is  of  theoretical  interest  as  an 
additional  proof  of  their  porosity.  It  has  been  found  that  a  pres 
sure  of  one  atmosphere  diminishes  the  volume  of  water  at  the 
freezing  point  by  yi^i^nr  5  ^"^  ^^^^  under  a  pressure  of  3000 
atmospheres,  its  volume  is  diminished  by  -}j^.  An  equal  pressure 
diminishes  the  volume  of  ether  by  about  J.  The  compressibility 
of  solids  not  having  sensible  pores  is  still  less  than  that  of  liquids. 
The  compressibility  of  glass,  for  example,  is  about  1^  as  great  as 
that  of  water. 

There  are  two  conceivable  explanations  of  compressibility; 
namely:  (i)  that,  by  compression,  the  molecules  of  a  body  are 
crowded  closer  together,  thus  diminishing  the  void  spaces  between 
them;  (2)  that  the  molecules  themselves  are  diminished  in  size  by 
compression.  If  the  latter  were  true,  matter  would  be  compress- 
ible even  if  it  were  not  porous.  However,  in  addition  to  the 
known  porosity  of  matter,  other  facts,  which  are  considered  in 
the  next  section,  indicate  that  the  first  of  the  above  suppositions 
is  the  true  explanation  of  compressibility. 

It  is  evident  from  the  very  great  compressibility  of  gases  that, 
under  ordinary  pressures,  their  molecules  occupy  only  an  exceed- 
ingly small  portion  of  the  space  allotted  to  them.  The  volume  of 
a  gram  of  steam  under  a  pressure  of  one  atmosphere  is  1661  ccm. 


Molecular  Motion  141 


—  a  volume  1661  times  greater  than  it  occupies  as  a  liquid. 
Even  if  we  were  to  assume  that,  as  a  liquid,  the  molecules  fill  the 
entire  space,  it  would  follow  that  the  average  distance  between 
the  molecules  of  steam,  under  a  pressure  of  one  atmosphere,  is 
nearly  twelve  times  the  diameter  of  a  molecule.  Oxygen  has  been 
subjected  to  a  pressure  of  3000  atmospheres,  under  which  pres- 
sure its  density  was  greater  than  that  of  water,  although  it  still 
remained  in  the  gaseous  state.  The  compressibility  of  all  gases, 
when  subjected  to  very  great  pressures,  is  less  than  that  indicated 
by  Boyle's  law,  showing  that  a  gas  approaches  the  condition  of  a 
liquid  as  its  molecules  are  crowded  together.  Why  the  molecules 
of  a  gas  remain  apart  under  enormous  pressures  is  a  question  con- 
sidered in  the  next  section. 
Laboratory  Exercise  26. 

179.  Expansibility  ;  Effect  of  Heat.  —  While  the  effect  of  pres- 
sure upon  the  volume  of  liquids  and  solids  is  far  too  small  to  be 
demonstrated  by  simple  experimental  methods,  the  effect  of  heat 
can  very  readily  be  shown,  as  illustrated  by  the  experiments  of  the 
preceding  laboratory  exercise.  In  general,  the  application  of  heat 
to  a  body  causes  it  to  expand,  and  with  a  loss  of  heat  it  contracts. 
With  liquids  and  solids  this  change  of  volume  is  slight ;  with  gases 
it  is  very  much  greater.  When  water  is  heated  from  the  freezing 
to  the  boiling  point,  it  increases  in  volume  by  four  per  cent. 
With  the  same  change  of  temperature,  the  pressure  remaining 
constant,  the  volume  of  a  gas  increases  over  thirty- six  per  cent. 
The  expansion  of  a  substance  by  heat  is  supposed  to  be  due  to  the 
wider  separation  of  its  molecules,  not  to  any  increase  in  their 
size. 

II.   Molecular  Motion 

180.  Diffusion  of  Gases.  —  When  two  or  more  gases  are  brought 
in  contact  and  left  undisturbed,  they  quickly  mix  with  one  another, 
even  in  cases  where  a  difference  of  density  would  tend  to  prevent 
such  mixing,  the  denser  gas  being  originally  at  the  bottom.     This 


142  Some  Properties  of  Matter 


spontaneous  mixing  of  gases  is  called  diffusion.    The  process  may 
be  illustrated  by  the  following  experiments. 

When  illuminating  gas  or  any  other  gas  having  a  strong  odor  is 
permitted  to  escape  into  a  room,  the  sense  of  smell  quickly  reveals 
the  presence  of  the  gas  in  all  parts  of  the  room.  This 
is  well  illustrated  by  a  little  ammonia  poured  on  the 
floor.  The  ammonia  evaporates,  and  the  odor  of 
ammonia  gas  fills  the  room  {Exp.). 

A  bottle  of  oxygen  and  another  of  illuminating  gas 
are  placed  together,  mouth  to  mouth  (Fig.  112),  the 
one  containing  the  oxygen  at  the  bottom.     After  they 
have  stood  thus  for  a  few  minutes  they  are  separated 
and  a  flame  is  quickly  applied  to  the  mouth  of  each. 
There   is   an   explosion   in   each  'case,  indicating   the 
presence  of  a  mixture  of  oxygen  and  illuminating  gas 
Fio.  na.    in  both  bottles.     Note  that,  since  the  denser  gas  (oxy- 
gen) is  placed  at  the  bottom,  the  mix- 
ing is  opposed  by  gravity  {Exp.). 

A  porous  cup  of  unglazed  earthen- 
ware is  fitted  with  a  cork  and  glass 
tube,  and  supported,  as  shown  in  Fig. 
113,  with  the  lower  end  of  the  tube  in 
a  tumbler  of  water.  A  large  jar  is  held 
over  the  cup  and  filled  with  illumi- 
nating gas  through  a  rubber  tube.  The 
experiment  has  three  stages.  ( i )  Bub- 
bles of  gas  immediately  rise  from  the 
end  of  the  glass  tube  in  the  water, 
indicating  that  the  pressure  within  the 
cup  has  been  increased  by  the  entrance 
of  some  of  the  gas  through  its  micro- 
scopic pores.  (2)  On  removing  the 
jar,  the  water  immediately  begins  to  rise  ^^°'  ^^3* 

in  the  tube,  indicating  a  decrease  of  pressure  within,  due  to  the 
escape  of  the  gas  through  the  pores.     (3)  Presently  the  water 


Molecular  Motion  143 

begins  to  fall  in  the  tube,  and  continues  to  do  so  until  the  level 
is  the  same  as  in  the  tumbler.  This  indicates  that  air  enters  the 
cup  through  the  pores  until  the  pressures  within  and  without  are 
equalized ;  and,  moreover,  suggests  that  the  fuller  explanation  of 
the  first  two  stages  of  the  experiment  is  as  follows :  Both  the 
illuminating  gas  and  the  air  can  diffuse  through  the  pores  of  the 
cup  ;  but  the  gas  more  rapidly  than  the  air.  In  (i)  the  pressure 
within  increases  because  the  gas  enters  more  rapidly  than  the  air 
escapes;  in  (2)  the  pressure  within  decreases  because  the  gas 
escapes  more  rapidly  than  the  air  can  enter. 

181.  Diffusion  explained  by  Molecular  Motion. — The  phe- 
nomena of  diffusion,  as  illustrated  by  the  above  experiments, 
indicate  that  the  molecules  of  a  gas  are  in  constant  motion.  In 
fact,  all  that  is  known  about  the  properties  and  laws  of  gases  sup- 
ports the  conclusion  that  the  molecules  of  a  gas  are  always  in  very 
rapid  motion,  darting  hither  and  thither  in  all  directions  at  random, 
like  bees  or  gnats  in  a  swarm.  Each  molecule,  according  to  the 
theory,  moves  in  a  straight  line  till  it  hits  another  molecule  or  the 
wall  of  the  containing  vessel,  when  it  is  reflected  and  bounds  off 
in  a  different  direction. 

When  two  gases  are  brought  in  contact,  the  molecules  of  each 
quickly  find  their  way  between  those  of  the  other,  and  in  a  short  time 
the  entire  space  is  filled  with  a  homogeneous  mixture  of  the  two. 
Molecular  motion  affords  an  explanation  of  diffusion  through  the 
porous  cup  in  the  last  experiment.  A  molecule  that  by  chance  is 
moving  toward  one  of  the  pores  finds  an  easy  entrance  and  a 
passageway  through  to  the  other  side.  It  must  not  be  supposed 
that  diffusion  ceases  when  equilibrium  of  pressure  is  established  at 
the  end  of  the  experiment.  With  the  same  gas  (air)  at  the  same 
temperature  and  pressure  on  both  sides  of  the  partition,  an  equal 
number  of  molecules  find  their  way  through  it  in  both  directions, 
thus  maintaining  the  equilibrium. 

If  two  gases  at  the  same  temperature  and  pressure  are  of 
unequal  density,  the  molecules  of  the  rarer  gas  are  moving 
more   rapidly  than   those    of  the  other.     This  accounts  for  the 


144  Some  Properties  of  Matter 

more  rapid  diffusion  of  the  illuminating  gas  through  the  porous 
cup. 

Diffusion  is  a  very  different  process  from  the  flow  of  gases  in 
currents.  In  diffusion,  the  molecules  move  as  individuals;  in 
currents,  they  move  collectively  as  one  body.  Currents  hasten 
the  process  of  mixing,  and  hence  aid  diffusion.  Diffusion  supple- 
ments the  action  of  winds  in  keeping  the  constituents  of  the  air 
uniformly  mixed. 

182.  Molecular  Motion  the  Cause  of  Gas  Pressure.  —  The  suppo- 
sition that  the  molecules  of  a  gas  are  in  motion  as  described  above 
affords  a  complete  explanation  of  gas  pressure  and  of  Boyle's  law. 
The  force  exerted  upon  the  wall  of  the  containing  vessel  when  a 
molecule  strikes  it  is  inappreciable ;  but  these  blows  are  so  numer- 
ous that  their  combinetl  effect  is  a  continuous,  constant  pressure. 
If  a  given  mass  of  gas  is  compressed  to  half  its  former  volume,  its 
density  is  doubled,  and  twice  as  many  molecules  strike  a  given 
portion  of  the  wall  of  the  containing  vessel  every  second  ;  hence 
the  pressure  is  doubled. 

Molecular  motion  accounts  also  for  the  indefinite  expansibility 
of  gases,  and  for  the  fact  that  the  molecules  remain  separated 
even  when  subjected  to  great  pressure. 

From  the  known  density  of  a  gas,  it  is  possible  to  compute 
what  the  average  velocity  of  its  molecules  must  be  in  order  to 
exert  the  observed  pressure.  The  computation  yields  the  sur- 
prising result  that  the  molecules  of  the  air  are  darting  about  with 
an  average  velocity  of  about  eighteen  miles  per  minute  (varying 
somewhat  with  the  temperature).  "Could  we,  by  any  means," 
says  Professor  Cooke,  "  turn  in  one  direction  the  actual  motion  of 
the  molecules  of  what  we  call  still  air,  it  would  become  at  once  a 
wind  blowing  seventeen  miles  per  minute,  and  would  exert  a  de- 
structive power  compared  with  which  the  most  violent  tornado  is 
feeble."  The  velocity  of  hydrogen  molecules  is  $.S  times  as  great 
as  that  of  air  molecules,  or  about  sixty-eight  miles  per  minute. 

183.  The  Kinetic  Theory  of  Gases. — The  explanation  of  the 
physicial  properties  of  gases  as  consequences  of  the  motion  of 


Molecular  Motion  145 

their  molecules  is  called  the  kinetic  theory  of  gases.  The  theory, 
in  its  complete  form,  applies  the  laws  of  dynamics  to  the  indi- 
vidual molecules,  and  accounts  definitely  {i.e.  quantitatively)  for 
all  the  laws  of  gases  ;  but  the  mathematical  treatment  of  the 
theory  belongs  to  advanced  physics. 

184.  The  Meaning  and  Value  of  a  Theory.  —  An  hypothesis,  or 
theory,  is  a  suggested  explanation  of  facts  that  cannot  be  traced 
to  any  directly  ascertainable  cause.  A  theory  is  not  necessarily 
true  even  if  it  affords  a  complete  explanation  of  all  the  known 
facts ;  for  it  is  conceivable  that  the  true  cause  may  be  very  differ- 
ent from  the  one  supposed.  In  fact,  it  has  happened  repeatedly 
in  the  history  of  science,  that  rival  theories  have  been  ably 
defended  at  the  same  time  by  different  scientists  of  recognized 
authority.  If,  at  any  time,  a  new  fact  is  discovered  which  is  in- 
consistent with  a  theory,  the  theory  must  be  modified  so  as  to 
be  in  agreement  with  the  fact,  or,  if  this  is  impossible,  it  must 
be  abandoned.  Newly  discovered  facts  have  often  served  to  dis- 
tinguish between  a  true  theory  and  a  false  one. 

Physical  theories  are  of  service  in  explaining  properties  of  matter, 
natural  laws,  and  natural  phenomena.  Thus,  as  we  have  seen,  the 
molecular  theory  of  matter  accounts  for  the  divisibility,  porosity, 
and  compressibility  of  matter;  and  the  kinetic  theory  of  gases 
fully  explains  the  phenomena  of  diffusion,  the  pressure  and  expan- 
sibility of  gases,  and  Boyle's  law. 

185.  Diffusion  of  Liquids.  —  If  any  two  liquids  that  can  be 
mixed  with  each  other  are  placed  in  the  same  vessel,  the  denser 
at  the  bottom,  and  left  undisturbed,  they  will  mix  by  diffusion, 
the  process  being  similar  to  the  diffusion  of  gases.  The  progress 
of  diffusion  in  liquids  is  visible  in  cases  where  it  is  accompanied 
by  a  change  of  color,  as  in  the  following  experiments. 

A  test  tube  or  a  tall  jar  is  nearly  filled  with  water  colored  with 
blue  litmus.  A  little  strong  sulphuric  acid  is  then  admitted  at  the 
bottom  through  a  long-stemmed  funnel  (Fig.  114).  The  acid  is 
considerably  denser  than  the  water  and  supports  it,  the  surface 
separating  the  two  being  distinctly  visible.     Since  acid  turns  blue 


146 


Some  Properties  of  Matter 


Fig.  114. 


litmus  red,  the  progressive  change  of  color  from  blue  to  red, 
which  slowly  takes  place  up  the  tube,  indicates  the  height  to 
which  the  acid  has  risen  by  diffusion  {Exp.). 

A  jar  is  half  filled  with  water  and  a  strong  solu- 
tion of  copper  sulphate  is  admitted  at  the  bottom,  as 
the  acid  is  in  the  preceding  experiment.  The  progress 
of  diffusion  is  indicated  by  the  very  slow  rise  of  the 
blue  color  of  the  solution.  The  process  requires 
months  for  its  completion  {Exp.), 

We  see  from  these  experiments  that  diffusion  takes 
place  in  litjuids,  as  in  gases^  without  the  aid  of  cur- 
rents and  in  opposition  to  gravity.  The  explanation 
is  therefore  the  same ;  the  molecules  of  a  liquid  are 
in  motion.  T^he  very  slow  rate  of  diffusion  in  liquids 
indicates  that  the  motion  of  the  molecules  is  greatly 
restricted  as  the  result  of  their  crowded  condition. 
The  molecules  of  a  liquid  are  always  moving  about  among  one 
another  whether  a  second  liquid  is  present  or  not ;  in  the  latter 
case,  however,  there  is  no  direct  evidence  of  this  motion. 

186.  Vibration  of  Molecules.  —  The  molecules  of  a  solid  are 
held  together  in  fixed  positions  ;  i.e.  they  have  no  motion  of  trans- 
lation. Many  facts,  however,  support  the  conclusion  that  the 
molecules  of  all  bodies,  including  solids,  are  in  a  state  of  rapid 
vibration.  We  may  think  of  the  vibratory  motion  of  a  molecule 
as  something  Hke  the  quivering  of  a  piece  of  jelly  when  it  is  shaken. 

The  molecules  of  a  gas  are  supposed  to  be  vibrating  freely  while 
moving  in  straight  lines  between  successive  collisions.  A  third 
motion  is  also  possible ;  namely,  rotation  —  like  the  spinning  of  a 
top.  In  liquids  these  motions  are  modified  and  restricted  by  the 
crowded  condition  of  the  molecules ;  in  solids  molecular  motion 
is  probably  restricted  to  an  irregular  vibration. 

187.  Theory  of  Heat.  —  According  to  the  kinetic  theory,  when 
a  gas  is  heated  the  velocity  of  its  molecules  is  increased.  This 
explains  the  expansion  of  a  gas  when  heated,  or,  if  expansion  is 
prevented,  the  increase  of  its  pressure.     Further  evidence  that 


Molecular  Forces  147 

heat  increases  molecular  motion  is  found  in  the  increased  rate  of 
diffusion,  both  of  gases  and  of  liquids,  at  higher  temperatures. 
The  expansion  of  liquids  aiid  solids  when  heated  is  also  attributed 
to  increase  of  molecular  motion.  The  more  energetic  vibration  of 
the  molecules  enables  them  to  push  their  neighbors  farther  away, 
and  the  whole  mass  expands  in  consequence. 

It  was  stated  in  Art.  148  that  heat  is  a  form  of  energy.  We  can 
now  understand  what  that  form  is.  According  to  the  theory  uni- 
versally accepted  for  more  than  half  a  century,  heat  is  the  kinetic 
energy  of  molecular  motion.  When  a  bullet  strikes  a  steel  target 
(Art.  148,  end),  the  molecules  of  the  bullet  and  of  the  adjacent 
parts  of  the  target  are  violently  disturbed  by  the  impact  and  set 
in  energetic  vibration.  The  energy  due  to  the  motion  of  the  bul- 
let as  a  whole  (molar  kinetic  energy)  is  t-ransformed  by  the  impact 
into  energy  due  to  the  motion  of  its  molecules  and  the  molecules 
of  the  target  (molecular  kinetic  energy  or  heat). 

III.    Molecular  Forces 

188.  Molecular  Attraction  and  Molecular  Pressure.  —  The  mole- 
cules of  a  body  are,  in  general,  subject  to  the  action  of  two  oppos- 
ing forces,  one  of  which  tends  to  bring  them  together,  and  the 
other  to  separate  them  from  each  other.  The  first  is  called  molecu- 
lar attraction,  or  cohesion.  It  is  strong  in  some  substances  and 
weak  in  others  ;  and,  in  the  same  substance,  it  grows  weaker  as 
the  distance  between  the  molecules  increases.  Its  action  is  very 
different  from  that  of  gravitation,  as  will  presently  be  seen. 

The  other  molecular  force  is  due  to  molecular  motion,  or  heat, 
and  consists  in  the  continuous  and  inconceivably  rapid  shower 
of  blows  which  each  molecule  exerts  upon  its  neighbors.  These 
blows  act  as  a  molecular  pressure,  the  tendency  of  which  is  to 
drive  the  molecules  farther  apart. 

189.  Molecular  Forces  and  the  States  of  Matter.  —  It  is  the 
mutual  relation  between  molecular  attraction  and  molecular  pres- 
sure due  to  heat  which  determines  whether  a  substance  will  be  in 


148  Some  Properties  of  Matter 

the  solid,  liquid,  or  gaseous  state.  In  solids  attraction  greatly  pre- 
dominates, and  the  molecules  hold  together,  or  cohere^  in  fixed 
relative  positions.  The  same  relation  holds  in  liquids  but  in  a  less 
degree ;  the  molecules  cohere,  but  are  able  to  wander  at  random 
among  one  another.  In  gases  at  ordinary  pressures  and  tempera- 
tures, the  molecules  are  so  widely  separated  that  attraction  is 
inappreciable  or  wholly  wanting,  and  molecular  pressure  is 
balanced  only  by  fxternai  force. 

When  the  molecular  motion  of  a  solid  is  sufficiently  increased 
by  heating,  cohesion  is  so  far  overcome  by  the  increased  molecu- 
lar pressure  that  the  solid  is  liquefied.  By  further  heating,  the 
molecular  i)ressure  may  become  so  great  as  to  separate  the  mole- 
cules beyond  the  range  of  their  mutual  attraction ;  the  liquid  will 
then  be  vaporized,  or  converted  into  a  gas. 

190.  Strength  of  Cohesion;  Tenacity. — To  break  a  body  or 
tear  it  apart,  a  force  must  be  exerted  sufficient  to  overcome  the 
cohesion  between  the  molecules  lying  on  opposite  sides  of  the 
surfaces  separated.  Thus  cohesion  is  the  cause  of  tenacity 
(the  property  of  a  body  inconsequence  of  which  it  resists  being 
pulled  or  broken  apart)  ;  and,  conversely,  the  tenacity  of  a  body 
serves  as  a  test  of  the  strength  of  cohesion  within  it. 

Steel  is  the  most  tenacious  of  all  substances,  a  tension  of  1 80 
lb.  being  required  to  break  a  steel  wire  having  a  cross-section 
of  I  sq.  mm.  For  an  equal  cross-section,  the  tenacity  of  soft 
copper  wire  is  about  70  lb.,  of  lead  wire  5  lb.,  of  glass  14  lb.,  of  oak 
(in  the  direction  of  its  fibers)  15  lb.  {Exp.). 

Laboratory  Exercise  j. 

191.  Effect  of  Distance  on  Cohesion.  —  When  the  pieces  of  a 
broken  stick  or  stone  are  accurately  fitted  and  firmly  pressed  to- 
gether, the  parts  do  not  cohere  with  appreciable  force  ;  for  the 
molecules  upon  the  two  surfaces  cannot  by  such  means  be  brought 
within  the  range  of  their  attractive  power.  Glue  or  cement,  spread 
between  the  surfaces,  bridges  the  gap,  and,  on  drying,  serves  as  an 
effective  connecting  link.  When  two  pieces  of  clean  plate  glass 
are  firmly  pressed  together,  they  cohere  with  sufficient  force  to 


Molecular   Forces  149 

sustain  the  weight  of  one  of  them.  The  pieces  are  not  held 
together  by  atmospheric  pressure,  as  the  Magdeburg  hemispheres 
are,  for  there  is  still  a  thin  layer  of  air  between  them,  and,  besides, 
the  experiment  succeeds  quite  as  well  under  the  receiver  of  an 
air  pump  from  which  the  air  has  been  exhausted.  Two  pieces  of 
common  window  glass  do  not  cohere  when  pressed  together.  The 
reason  for  the  difference  is  that  the  surfaces  of  the  plate  glass  are 
quite  accurately  plane,  and  hence  fit  very  closely  together  over 
their  whole  extent ;  while  the  surfaces  of  common  window  glass 
are  more  or  less  wavy,  and  really  come  in  contact  at  only  a  few 
points.  Two  pieces  of  any  metal,  having  surfaces  accurately  plane, 
cohere  like  the  pieces  of  plate  glass. 

These  experiments  afford  some  indication  of  the  extremely  short 
distance  to  which  cohesion  can  act.  It  is  estimated  that  this  dis- 
tance is  not  greater  than  ^u-ffTny  ''"^^-  —  about  y^Vir  ^^  ^^^  thick- 
ness of  the  paper  on  which  this  is  printed.  Cohesion  is  weaker 
between  two  pieces  of  plate  glass  than  it  is  within  them  only 
because  the  distance  between  the  attracting  molecules  is  greater. 

192.  Cohesion  of  Plastic  Solids ;  Welding.  —  Two  surfaces  of  a 
plastic  solid,  as  soft  putty  or  clay,  warm  molasses  candy,  or  wax, 
can  be  brought  so  close  together  by  pressure  that  they  will  unite 
perfectly.  Even  two  pieces  of  lead  will  cohere  quite  firmly  if 
their  freshly   brightened  surfaces  are  pressed  together  in  a  vise 

In  the  process  of  welding,  the  two  pieces  of  metal  to  be  united 
are  rendered  plastic  by  heating.  Their  surfaces  are  then  brought 
within  the  range  of  cohesion  by  hammering  them  together.  Simi- 
larly, two  pieces  of  glass,  rendered  plastic  by  heating,  will  readily 
unite,  forming  perfect  union  (£x/>.). 

193.  Adhesion.  —  There  is  no  evidence  of  any  difference  in  the 
nature  of  the  attraction  between  molecules  of  the  same  kind  and 
that  between  molecules  of  different  kinds;  but  the  former  is 
generally  called  cohesion,  and  the  latter  adhesion.  The  distinction 
is  convenient  but  unimportant. 

There  are  many  familiar  examples  of  adhesion  between  solids. 


1 50  Some  Properties  of  Matter 

Mud  sticks,  or  adheres,  to  any  object  with  vexing  facility ;  butter 
adheres  to  a  knife  and  to  the  bread  upon  which  it  is  spread.  The 
adhesion  of  metals  is  utilized  in  gold  and  silver  plating.  Ordi- 
narily there  is  no  appreciable  adhesion  between  solids  when  brought 
in  contact,  unless  one  of  them  is  plastic ;  but  this  is  only  because 
their  surfaces  are  not  brought  sufficiently  close  together. 

Adhesion  between  solids  and  liquids  is  also  familiar.  In  a  large 
majority  of  cases,  when  a  liquid  and  a  solid  are  brought  in  contact, 
the  liquid  clings  to  the  solid  and  wets  it.  This  is  because  adhe- 
sion between  the  two  is  greater  than  cohesion  within  the  liquid. 
Thus  when  the  finger  is  dipped  in  water  and  removed,  the  layer 
of  water  in  contact  with  the  finger  is  held  by  adhesion  with  suffi- 
cient force  to  tear  it  away  from  the  adjacent  molecules  of  water. 
In  some  cases  a  solid  is  not  wet  by  a  liquid.  Water,  for  example, 
does  not  wet  a  surface  covered  with  grease  or  wax,  and  mercury 
wets  but  few  substances.  In  such  cases  the  liquid  clings  together 
in  somewhat  flattened  drops  upon  the  surface  of  the  solid  (Fig. 
115).  This  behavior  of  the  liquid  does 
not  prove  that  adhesion  is  wanting.  On 
the  contrary,  there  is  direct  experimental 
iG.  115.  evidence  of  adhesion ;  and,  between  gla§s 

and  mercury,  it  is  even  very  considerable  (Laboratory  Ex.  3). 
The  explanation  is  that,  in  such  cases,  cohesion  in  the  liquid  is 
stronger  than  adhesion,  whatever  the  strength  of  the  latter  may  be. 
Gases  also  adhere  to  solids,  forming  a  very  thin  but  dense  layer 
upon  their  surface.  In  setting  up  a  barometer,  air  adheres  to  the 
inner  wall  of  the  tube,  and  is  driven  off  only  by  heating  the  mer- 
cury till  it  boils.  The  small  bubbles  of  air  that  gather  upon  the 
side  of  a  glass  of  water  as  it  becomes  warm  are  held  there  by 
adhesion  with  sufficient  force  to  overcome  the  buoyancy  of  the 
water. 

194.  Cohesion  and  Gravitation  Compared. —  We  know  the  law 
of  gravitation,  but  not  its  cause  (Art.  1 29).  We  know  neither  the 
law  nor  the  cause  of  cohesion ;  but  it  is  evident  that  the  law  is 
very  different  from  that  of  gravitation,  for  cohesion  acts  only  at 


Surface  Tension  and   Capillarity        151 

insensible  distances,  and  within  such  distances  it  is  enormously 
stronger  than  gravitation  between  the  same  masses.  Hence  the 
strength  of  bodies  in  general  depends  practically  entirely  on  cohe- 
sion, and  gravitation  becomes  appreciable  only  in  bodies  of  very 
great  size.  The  strength  of  the  earth  depends  almost  wholly  on 
gravitation.  If  we  suppose  the  earth  to  be  divided  into  hemi- 
spheres by  any  plane  through  its  center,  the  gravitational  attraction 
by  which  the  hemispheres  are  held  together  is  one  hundred  times 
as  great  as  cohesion  would  be  if  the  earth  were  made  of  solid 
steel.  If  there  were  a  planet  fifty  miles  in  diameter  having  the 
same  density  as  the  earth  and  the  tenacity  of  sandstone,  gravi- 
tation and  cohesion  would  be  equally  effective  in  keeping  it 
together. 


IV.   Surface  Tension  and  Capillarity 

195.  Surface  Tension.  —  A  pin  or  a  needle  can  be  made  to  float 
on  water,  if  carefully  laid  upon  the  surface  so  that  neither  end 
touches  the  water  before  the  other.  If  the  pin  sinks  and  becomes 
wet,  it  should  be  dried  by  rubbing  it  between  the  fingers  or  in 
the  hair,  as  this  covers  it  with  a  coating  of  oil  and  diminishes  the 
adhesion  of  the  water  to  it  {Exp.).  The  floating  pin  lies  in  a 
depression  or  trough  which  is  several  times  larger  than  itself 
(represented  in  cross-section  in  Fig.  116).  It  is,  in  fact,  sup- 
ported in  much  the  same  way  as  a  person  is 
when  lying  in  a  hammock ;  the  cords  of  the 
hammock  are  under  tension,  and  this  tension,  ?^3§^^§Q 
acting  obliquely  upward  on  all  sides,  constitutes  ~-^-f=  :^^^r^£3-iEf 
the  supporting  force.     Similarly,  the  floating  of  ^^^'  ^^^• 

the  pin  is  explained  by  supposing  the  surface  of  the  water  to  be 
somewhat  tough  and  in  a  state  of  tension. 

The  properties  of  the  surfaces  of  liquids  can  be  studied  to 
advantage  by  means  of  liquid  films.  It  can  be  shown  by  a  num- 
ber of  simple  experiments  that  soap  films  and  soap  bubbles  are  in 


152  Some  Properties  of  Matter 

a  state  of  tension^  (Laboratory  Exercise  4).  A  soap  bubble  is 
spherical,  like  a  toy  balloon,  because  the  film  of  the  bubble  is  in  a 
state  of  uniform  tension  throughout,  just  as  the  rubber  of  the  bal- 
loon is.  Since  a  spherical  surface  is  smaller  than  any  other  that 
incloses  an  equal  volume,  the  film  of  a  bubble  assumes  this  shape 
in  shrinking  as  much  as  possible. 

A  drop  of  any  liquid,  when  freed  from  the  distorting  effect  of 
its  weight,  as  in  falling,  is  spherical.  A  drop  of  oil  suspended  in  a 
solution  of  alcohol  and  water  of  its  own  density  is  an  excellent 
illustration  (Exp,).  The  spherical  form  of  a  drop  is  due  neither 
to  the  mutual  gravitation  of  its  particles  nor  to  cohesion  acting 
throughout  its  mass,  but  to  the  tension  of  its  surface.  Its  surface, 
like  the  film  of  a  bubble,  contracts  as  much  as  possible,  and  when 
thus  contracted  is  spherical. 

Laboratory  Exercise  4. 

196.  Cause  of  Surface  Tension.  —  Each  molecule  of  a  liquid 
has  hundreds,  perhaps  thousands,  of  neighbors  near  enough  to 
attract  it  by  cohesion,  all  of  which  are  inclosed  within  a  spheri- 
cal surface  having  the  given  molecule  as  a  center  and  a  radius 
equal  to  the  greatest  distance  to  which  cohesion  can  act  (a  dis- 
tance of  less  than  microscopic  magnitude).  Any  molecule  of  a 
liquid  whose  distance  from  the  surface  is  greater  than  the  radius 
of  such  a  sphere  is  attracted  equally  in  all  directions,  since  the 
molecules  within  the  range  of  cohesion  are  uniformly  disturbed 
around  it.  A  molecule  at  the  surface,  however,  is  attracted  inward, 
but  not  outward  ;  and  any  molecule  whose  distance  from  the  sur- 
face is  less  than  the  range  of  cohesion  is  more  strongly  attracted 
inward  than  outward,  since  the  greater  number  of  the  molecules 
that  are  near  enough  to  attract  it  lie  on  the  inside.  The  result  is 
that  the  molecules  at  the  free  surface  of  a  liquid  exert  a  strong 
inward  pressure,  which  tends  to  reduce  the  surface  to  the  least 
possible  area,  the  effect  being  the  same  as  if  the  liquid  were 

1  Recipe  for  a  good  soap  solution :  Put  2  oz.  of  Castile  or  palm-oil  soap,  shaved 
thin,  in  i  pt.  of  distilled  or  rain  water.  Shake,  pour  off  the  clear  solution,  add  to  it 
\  pt.  of  glycerine,  and  stir. 


Surface  Tension  and  Capillarity        153 

inclosed  in  a  stretched,  elastic  membrane.  Thus  the  surface 
tension  of  a  liquid  is  due  to  the  action  of  cohesion  at  its 
surface. 

197.  Surface  Tension  of  Different  Liquids. — The  surface  ten- 
sion of  different  liquids  has  been  determined  by  experiment,  and 
it  has  been  found  to  be  greater  for  water  than  for  any  other  liquid 
except  mercury ;  hence  the  surface  tension  of  water  is  diminished 
by  mixing  any  other  substance  with  it.  This  is  readily  illustrated 
by  placing  a  drop  of  alcohol,  ether,  or  oil  on  the  surface  of  a  tum- 
bler of  water  beside  a  floating  sliver  of  wood.  The  bit  of  wood  is 
quickly  jerked  away  from  the  drop  by  the  greater  tension  of  the 
pure  water  on  the  other  side  {Exp.). 

198.  Surface  Viscosity.  —  The  surface  of  most  liquids  becomes 
more  viscous  than  the  interior  mass  after  exposure  to  the  air 
for  some  time ;  and  it  is  to  this  superficial  viscosity,  rather 
than  to  surface  tension,  that  the  strength  of  a  liquid  film  is  due. 
Only  very  small  bubbles  can  be  formed  on  the  surface  of  pure 
water,  and  these  quickly  break,  for  the  particles  rapidly  slip  away 
from  the  highest  part  of  the  bubble,  leaving  it  too  thin  to  hold. 
The  particles  in  a  soap  film  move  much  more  sluggishly,  on 
account  of  their  greater  viscosity ;  hence  a  much  longer  time 
elapses  before  any  part  of  the  film  becomes  so  thin  as  to  break, 
although  'the  surface  tension  of  the  film  is  less  than  that  of  pure 
water. 

199.  Capillarity.  —  The  combined  action  of  surface  tension  and 
adhesion,  when  a  liquid  and  a  solid  are  in  contact,  gives  rise  to  a 
class  of  phenomena,  called  capillary  pheti07?iena  because  they  are 
most  conspicuous  in  tubes  of  small  bore  (Latin  capilliis,  a  hair). 
Capillary  action  and  capillarity  are  general  terms  for  such 
phenomena. 

There  are  two  types  of  cases  to  be  considered,  represented 
respectively  by  water  and  mercury,  each  in  contact  with  glass. 
The  surface  of  water  in  a  clean  glass  vessel  is  turned  sharply 
up  at  the  edge  in  a  smooth  curve  (Fig.  117);  while  the  sur- 
face  of  mercury  is  curved  downward   at   the   edge  (Fig.    118) 


154  Some  Properties  of  Matter 

(£x/.).  In  all  cases  where  adhesion  exceeds  cohesion  in  the 
liquid  (cases  in  which  the  liquid  wets  the  solid),  the  edge  of  the 
liquid  is  drawn  up  against  the  surface  of  the  solid ;  in  all  cases 
where  cohesion  in  the  liquid  exceeds  adhesion  (cases  in  which  the 
solid  is  not  wet  by  the  liquid),  the  edge  of  the  liquid  is  draw*?f'' 
inward  and  away  from  the  surface  of  the  solid. 

When  small  glass  tubes  of  different  bore  are  held  vertically  in 
water,  the  water  rises  in  each  above  the  level  in  the  vessel,  and 
stands  higher  in  the  tube  of  smaller  bore  {Exp.),    The  surface 


i  (f)  if 

mill 


Fig.  117. 


Fu;.  lid. 


of  the  water  in  the  tube  is  concave  (viewed  from  above),  and  is 
curved  throughout,  forming  a  hemisphere,  if  the  tube  is  not  too 
large.  This  curved  surface,  like  that  of  a  soap  bubble,  exerts  a 
pressure  on  the  concave  side,  and  thus  partly  sustains  the  pressure 
of  the  air  upon  it.  The  water  therefore  rises  in  the  tube  till  equi- 
librium is  restored.  Atmospheric  pressure,  however,  is  not  essential 
to  capillary  action.  The  water  would  stand  at  the  same  level  in 
the  tubes  if  the  vessel  and  contents  were  placed  in  a  vacuum  ;  for 
the  concave  surface  in  the  tube  would  exert  the  same  upward  force 
as  before,  and  would  rise,  carrying  the  column  with  it  by  cohesion. 
The  greater  elevation  of  the  water  in  the  smaller  tube  is  due  to 


Surface  Tension  and  Capillarity        155 

the  fact  that  the  curvature  of  its  surface  is  greater  (radius  of  curva- 
ture less)  ;  for  it  can  be  proved  mathematically  that  the  pressure 
(per  unit  area)  exerted  toward  the  concave  side  by  a  curved  sur- 
face under  a  given  tension  is  inversely  proportional  to  the  radius 
of  curvature  of  the  surface.  Thus  if  the  diameter  of  the  smaller 
tube  is  one  half  that  of  the  larger,  the  upward  pressure  exerted  by 
the  water  surface  in  it  will  be  twice  as  great  and  will  support  a 
column  of  water  twice  as  high  as  that  in  the  larger. 

Mercury  stands  at  a  lower  level  in  capillary  tubes  than  it  does  in 
the  containing  vessel,  and  the  surface  in  the  tubes  is  convex  (Fig. 
118)  {Exp.).  The  pressure  exerted  by  the  curved  surface  is 
toward  the  concave  side,  as  with  water ;  but  it  causes  depression 
in  this  case,  as  the  concave  side  is  downward.  The  depression  is 
inversely  proportional  to  the  diameier  of  the  tube,  for  the  reasons 
given  in  the  case  of  capillary  elevation. 

Experiments  with  other  liquids  and  tubes  of  other  materials 
would  yield  results  agreeing  with  the  cases  considered,  as  expressed 
in  the  following  laws  :  — 

I.  If  a  liquid  wets  a  capillary  tube,  its  surface  is  concave  and 
it  is  drawn  up  ;  if  it  does  not  wet  the  tube,  its  surface  is  convex  and 
it  is  depressed. 

II.  The  elevation  or  the  depression  in  a  capillary  tube  is  in- 
versely proportional  to  the  diameter  of  the  tube. 

200.  Illustrations  of  Capillary  Action.  —  The  sensible  pores  of 
solids  serve  as  capillary  tubes  in  absorbing  liquids.  The  absorp- 
tion of  water  by  a  sponge,  of  ink  by  blotting  paper,  and  of  coffee 
by  a  lump  of  sugar,  are  familiar  examples.  The  flame  of  a  lamp  is 
fed  by  oil  which  is  drawn  up  through  the  wick  by  capillary  action. 
In  dry  weather  the  moisture  is  drawn  up  from  a  depth  of  many 
feet  through  the  pores  in  the  soil,  and  evaporates  at  the  surface. 
Cultivation  of  the  soil  increases  the  size  of  the  pores,  and  conse- 
quently checks  the  rise  of  water  through  the  cultivated  layer,  thus 
diminishing  the  loss  by  evaporation  at  the  surface. 

Capillary  action  would  enable  a  short  siphon  having  a  capillary 
bore  to  work  in  a  vacuum. 


156  Some  Properties  of  Matter 

PROBLEMS 

1.  What  is  the  distinction  between  a  theory  and  a  law?    between  a  theory 
and  a  fact  ?     Is  a  law  a  fact  ? 

2.  What  is  the  essential  difference  between  a  law  of  nature  and  a  law 
enacted  by  a  legislative  body? 

3.  If  a  gas  is  heated  but  not  permitted  to  expand,  how  is  the  pressure  that 
it  exerts  affected  ?     Explain. 

4.  Capillary  phenomena  are  sometimes  said  to  be  due  to  "  capillary  at- 
traction."    What  b  capillary  attraction? 


V.  Properties  due  to  Molecular  Forces 

201.  Molecular  forces  give  rise  to  different  specific  properties 
in  different  substances.  One  of  these,  tenacity,  has  already  been 
considered  (Art.  190).  The  list  includes  also  elasticity,  plasticity, 
brittleness,  malleability,-  ductility,  and  viscosity. 

202.  Elasticity.  —  Elasticity  is  the  property  of  matter  in  virtue 
of  which  bodies  resume  their  original  form  or  volume  when  any 
force  that  has  altered  their  form  or  volume  is  removed. 

Elasticity  of  volume  is  shown  by  recovery  of  volume  after 
compression.  It  is  a  general  property  of  matter  and  is  due  to 
molecular  pressure.  All  fluids  have  perfect  elasticity  of  volume  ; 
/>.  however  great  the  compression,  they  always  expand  to  their 
original  volume  when  the  pressure  is  removed.  Solids,  however, 
may  be  permanently  diminished  in  volume  to  a  slight  extent  by 
the  application  of  sufficient  pressure.  The  rolling  and  stamping 
to  which  silver  is  subjected  in  the  process  of  coining  causes  a 
decrease  of  volume  amounting  to  about  four  per  cent. 

Elasticity  of  form  is  a  specific  property  possessed  only  by  cer- 
tain solids.  It  is  shown  by  recovery  of  form  after  distortion.  An 
elastic  solid  is  able  to  recover  from  distortion  whether  caused  by 
compression,  stretching,  twisting,  or  bending.  Rubber  is  a  fa- 
miliar example.  Solids,  such  as  soft  putty  or  clay,  which  have  no 
power  to  recover  from  distortion,  are  called  inelastic  or  plastic. 

203.  Limit  of  Perfect  Elasticity.  —  An  elastic  solid  has  perfect 
elasticity  only  for  distortion  within  a  certain  limit,  called  the  limit 


Properties  due  to  Molecular  Forces      157 

of  perfect  elasticity,  or  generally,  the  limit  of  elasticity.  When  a 
body  is  distorted  beyond  its  limit  of  elasticity,  it  either  breaks  or 
takes  a  permanent  set,  i.e:  undergoes  a  permanent  change  of  form 
{Exp.).  In  the  first  case  the  solid  is  said  to  be  brittle;  in  the 
second  case  tough,  malleable,  or  ductile,  according  to  the  manner 
in  which  its  form  can  be  changed  (Art.  207).  It  is  a  mistake  to 
regard  brittle  substances  as  inelastic.  A  long,  slender  strip  of 
glass  or  piece  of  glass  tubing  is  quite  flexible  and  springs  back  to 
its  former  shape  when  released  {Exp.)  ;  within  its  limit  of  elas- 
ticity glass  is  perfectly  elastic. 

Substances  differ  widely  in  their  limits  of  elasticity.  A  piece  of 
soft  copper  wire  takes  a  permanent  set  before  it  is  bent  far,  —  its 
limit  of  elasticity  is  small.  A  piece  of  steel  wire  can  be  bent  many 
times  farther  without  a  permanent  change  of  form  {Exp.).  Rub- 
ber is  remarkable  for  its  large  limit  of  elasticity.  There  is  no 
limit  to  the  elasticity  of  volume  of  fluids.  However  great  the 
pressure  to  which  a  fluid  may  be  subjected,  recovery  of  volume 
is  always  complete  when  the  pressure  is  removed. 

204.  Elastic  Force ;  Measure  of  Elasticity.  —  In  common 
speech  we  say  that  a  body  is  very  elastic  or  highly  elastic  if  its 
limit  of  elasticity  is  large  (as  rubber),  or  if  it  is  highly  compress- 
ible (as  gases).  In  scientific  usage  these  terms  have  an  alto- 
gether different  meaning.  The  elastic  force  of  a  body,  or  the 
force  with  which  it  tends  to  recover  from  compression  or  distor- 
tion, is  the  measure  of  its  elasticity. 

The  elasticity  of  a  gas  is  measured  by  the  pressure  (per  unit 
area)  which  it  exerts  upon  the  sides  t>f  the  containing  vessel,  and 
is  increased  by  compression  (Art.  44).  The  elasticity  of  liquids 
is  much  greater  than  that  of  gases,  and  the  elasticity  of  ivory, 
glass,  or  steel  is  very  great  compared  with  that  of  rubber. 

205.  Elastic  Impact.  —  When  an  elastic  sphere  (rubber,  ivory, 
steel,  or  hardwood  ball,  or  glass  marble)  is  dropped  upon  a  smooth, 
rigid  surface,  as  a  large,  flat  plate  of  stone  or  steel,  it  rebounds 
nearly  to  the  height  from  which  it  was  dropped  {Exp.).  The 
rebound  is  explained  as  follows  :  The  force  of  the  impact  flattens 


158  Some  Properties  of  Matter 

the  ball  very  slightly  before  it  is  stopped  ;  but,  being  elastic,  it 
instantly  recovers  its  form,  and,  in  doing  so,  //  continues  to  press 
against  the  plate^  much  as  a  boy  pushes  against  the  ground  in  the 
act  of  jumping,  and  with  a  similar  result. 

Although  the  ball  recovers  its  form  completely,  the  force  with 
which  it  does  so  is  in  all  cases  somewhat  less  than  the  distorting 
force.  If  the  recovery  of  force  \yere  complete,  the  ball  would 
rebound  to  the  height  from  which  it  was  dropped,  but  this  is 
impossible,  as  some  of  the  energy  of  the  ball  is  transformed  by  the 
impact  into  heat  and  sound.  The  experiment  proves  the  elas- 
ticity of  bodies  that  are  often  regarded  as  perfectly  rigid.  More- 
over, by  this  test,  the  elasticity  of  glass  is  much  more  nearly 
perfect  than  that  of  nibber.     (How  shown?) 

206.  Stress  and  Strain.  —  Any  force  or  combination  of  forces 
tending  to  change  the  shape  or  size  of  a  body  is  called  a  stress. 
A  rubber  band  is  stretched  by  two  equal  and  opposite  pulls,  con- 
stituting a  tensiie  stress.  A  tensile  stress  causes  elongation.  A 
stress  consisting  of  equal  and  opposite  pressures  causes  compres- 
sion. Any  change  of  shape  or  size  of  a  body,  especially  of  a 
solid,  produced  by  the  action  of  a  stress,  is  called  a  strain.  A 
body  in  which  a  stress  produces  no  appreciable  strain  is  called 
rigid ;  but  no  body  is  perfectly  rigid,  i.e.  absolutely  unyielding  to 
stress.  In  the  study  of  the  mechanics  of  solids  we  have  disre- 
garded the  strains  produced  by  the  forces  under  consideration. 

207.  Plasticity;  Malleability  and  Ductility.  —  The  specific 
properties  of  bodies  have  no  sharply  defined  limits ;  on  the  con- 
trary, they  merge  insensibly  into  one  another  in  numerous  instances, 
and  the  names  of  such  properties  have,  in  consequence,  a  variable 
meaning.  This  is  the  case  with  the  term  plasticity,  which,  in  the 
customary  and  narrow  sense,  is  applied  to  bodies  that  can  be 
molded  by  moderate  pressure  into  any  desired  form.  Soft  putty 
and  clay  are  typical  examples.  But  gold  and  silver  exhibit  es- 
sentially the  same  property  under  much  greater  pressure,  but  at 
ordinary  temperatures,  when  they  receive  the  impression  of  the 
die  or  mold  in  stamping  coins ;  and  are  therefore,  in  a  broader 


Properties  due  to  Molecular  Forces      159 

sense,  called  plastic.  In  this  sense  all  elastic  bodies  that  are  not 
brittle  are  plastic  beyond  their  limit  of  elasticity.  This  form  of 
plasticity  is,  however,  generally  called  malleability  or  ductility. 

A  substance  is  said  to  be  malleable  if  it  can  be  hammered  or 
rolled  into  sheets ;  duciiky  if  it  can  be  drawn  out  into  the  form  of 
a  wire.  In  both  cases  ordinary  temperatures  are  to  be  understood 
unless  otherwise  stated.  Thus  we  should  say  that  glass  is  brittle, 
not  ductile ;  but  glass  is  very  ductile  when  heated  to  redness 
{Exp^,  Gold,  silver,  and  platinum  are  the  most  ductile  metals, 
gold  the  most  malleable.  Gold  has  been  beaten  into  leaves  so 
thin  that  six  hundred  of  them  placed  one  upon  another,  would 
have  a  thickness  no  greater  than  the  paper  upon  which  this  is 
printed. 

208.  Viscosity  and  Mobility.  —  A  substance  is  said  to  be 
viscous  if  its  form  changes  more  or  less  slowly  under  the  action  of 
its  own  weight.  Shoemaker's  wax,  pitch,  and  molasses  candy 
slightly  warmed  are  examples  of  viscous  solids.  Such  substances 
are  sometimes  classed  as  liquids,  since,  when  unsupported  at  the 
sides,  they  slowly  flatten  out,  or  flow.  Molasses  and  honey  are 
examples  of  viscous  liquids.  The  term  is  commonly  applied 
only  to  those  liquids  that  possess  the  property  in  a  considerable 
degree,  and  liquids  that  flow  readily,  as  water  and  alcohol,  are 
called  mobile.  But  all  liquids  and  gases  possess  viscosity  in 
some  degree. 

Viscosity  is  that  property,  due  to  internal  or  molecular  fric- 
tion, in  virtue  of  which  liquids,  gases,  and  some  solids  offer  re- 
sistance to  an  instantaneous  change  of  their  shape  or  of  the 
arrangement  of  their  parts,  although  they  offer  no  permanent  re- 
sistance to  such  change.  The  viscosity  of  water  is  shown  by  the 
behavior  of  a  bowl  or  tumbler  of  water  that  has  been  stirred  till 
it  is  rapidly  rotating.  If  the  motion  of  the  different  portions  of 
the  water  is  made  visible  by  means  of  sawdust  or  other  floating 
particles,  it  will  be  found  that  the  rate  of  rotation  steadily  de- 
creases, and  at  any  instant  is  greatest  at  the  center  and  least  at 
the  sides  of  the  vessel  (^Exp.),    The  explanation  is  that  friction 


i6o  Some  Properties  of  Matter 

against  the  sides  of  the  vessel  retards  the  layer  of  water  in 
contact  with  it,  and  this  layer  in  turn  retards  the  more  rapidly 
moving  layer  next  within.  This  retarding  action — due  to  inter- 
nal friction,  as  adjacent  layers  slip  over  one  another  in  their  un- 
equal motion — extends  throughout  the  entire  mass,  and  finally 
brings  the  whole  to  rest. 

The  dividing  line  between  viscous  liquids  and  plastic  solids  is 
not  clearly  defined ;  the  two  classes  merge  insensibly  into  each 
other.  Molasses  candy,  in  cooling,  passes  continuously  from  the 
stale  of  a  viscous  liquid  to  that  of  a  plastic  solid ;  and,  on  cooling 
further,  becomes  hard  and  brittle. 

209.  Hardness.  —  A  body  is  said  to  be  harder  than  another 
if  it  can  be  used  to  scratch  the  latter  but  cannot  be  scratched  by 
it.  Diamond  is  the  hardest  substance  known.  Pure  metals  are 
softer  than  their  alloys;  hence  gold  and  silver  used  for  money 
and  jewelry  are  alloyed  with  copper  to  increase  their  hardness. 

Steel  is  made  very  hard  by  sudden  cooling  after  it  has  been 
raised  to  a  high  temperature.  The  process  is  called  tempering. 
All  cutting  instruments  are  made  of  tempered  steel. 

PROBLEMS 

1.  The  top  of  a  river  flows  faster  than  the  bottom,  and  the  middle  flows 
faster  than  the  sides.     Why? 

2.  Write  a  list  of  all  the  general  projierties  of  matter  that  have  been  con- 
sidered in  this  or  in  previous  chapters.  Write  a  similar  list  of  specific  prop- 
erties. 


CHAPTER  VIII 

HEAT 

I.  Heat  and  Temperature 

210.  The  Caloric  Theory.  —  Before  the  final  acceptance  of  the 
present  theory  of  heat  (Art.  187)  about  the  middle  of  the  last 
century,  heat  was  very  generally  supposed  to  be  an  invisible  fluid 
without  weight,  which  could  of  itself  pass  from  a  hotter  to  a  colder 
body.  This  supposed  fluid  was  called  caloric  (Latin  calor^  heat). 
With  the  overthrow  of  the  theory,  the  word  itself  has  become 
obsolete  ;  but  the  root  occurs  in  several  words  in  common  use, 
among  which  are  calorie,  one  of  the  heat  units ;  caloritnetry,  the 
art  or  process  of  measuring  heat ;  and  calorimeter,  a  vessel  in 
which  substances  are  placed  in  measuring  their  gain  or  loss  of 
heat.  The  use  of  these  words  does  not  imply  any  reference  to 
the  caloric  theory;  but,  unfortunately,  the  terms  latetit  heat 
and  radiant  heat,  which  also  owe  their  origin  to  the  caloric 
theory,  are  distinctly  misleading,  for  neither  is  heat  at  all. 

211.  The  Mechanical  Theory  of  Heat:  Historical. —  "  The  first 
prominent  physicist  who  endeavored  to  overthrow  the  caloric 
theory  of  heat  was  Benjamin  Thompson"  (1753-1814),  a  native 
of  Massachusetts.  He  is  better  known  as  Count  Rumford,  having 
received  that  title  from  the  Elector  of  Bavaria,  whose  service  he 
entered  while  still  a  young  man.  While  engaged  in  the  boring 
of  cannon  for  the  Bavarian  government,  he  was  surprised  at  the 
heat  generated.  "  Whence  comes  this  heat  ?  What  is  its  nature  ? 
He  arranged  apparatus  so  that  the  heat  generated  by  the  friction 
of  a  blunt  steel  borer  raised  the  temperature  of  a  quantity  of 
water.     In  his  third  experiment,  water  rose  in  one  hour  to  107° 

161 


1 62  Heat 

Fahrenheit;  in  one  hour  and  a  half  to  142° ;  *at  the  end  of  two 
hours  and  thirty  minutes  the  water  actually  boiled.'  *  It  is  difficult 
to  describe  the  surprise  and  astonishment,'  says  Rumford,  '  ex- 
pressed in  the  countenances  of  the  bystanders,  on  seeing  so  large 
a  quantity  of  cold  water  (i8|  lb.)  heated,  and  actually  made  to 
boil  without  any  fire.'  .  .  .  The  source  of  heat  generated  by  fric- 
tion *  appeared  evidently  to  be  inexhaustible.'  The  reasoning 
by  which  he  concluded  that  heat  was  not  matter,  but  was  due  to 
motion,  we  can  give  only  in  part.  He  says,  *  It  is  hardly  neces- 
sary to  add  that  anything  which  any  insulated  body,  or  system  of 
bodies,  can  continue  to  furnish  without  limitation,  cannot  possibly 
be  a  material  substance  ;  and  it  appears  to  me  extremely  difficult, 
if  not  quite  impossible,  to  form  any  distinct  idea  of  anything  ca- 
pable of  being  excited  and  communicated  in  the  manner  in  which 
heat  was  excited  and  communicated  in  these  experiments,  except 
it  be  motion.* 

"  Rumford 's  conclusion  regarding  the  nature  of  heat  was  vigor- 
ously attacked  by  the  calorists,  but  it  was  confirmed  in  1 799  by 
Sir  Humphry  Davy.  By  means  of  clockwork  he  rubbed  two 
pieces  of  ice  against  one  another  in  the  vacuum  of  an  air  pump. 
Part  of  the  ice  was  melted,  although  the  temperature  of  the  re- 
ceiver was  kept  below  the  freezing  point.  From  this  he  con- 
cluded that  friction  causes  vibration  of  the  corpuscles  of  bodies, 
and  this  vibration  is  heat."  —  Cajori's  History  of  Physics. 

Notwithstanding  the  work  of  Rumford  and  Davy,  but  few 
physicists  were  convinced.  In  fact,  the  caloric  theory  was  not 
completely  discredited  until  after  J.  P.  Joule,  an  Englishman, 
had  proved,  by  a  series  of  experiments  extending  over  a  period 
of  ten  years  (1840  to  1850),  that  the  amount  of  heat  which  can  be 
generated  by  a  given  amount  of  mechanical  work  is  invariable 
(Art.  270). 

212.  Sources  of  Heat.  —  All  other  forms  of  energy  are  capable 
of  transformation  into  heat.  Mechanical  energy  is  transformed 
into  heat  by  friction  and  impact,  as  we  have  already  seen,  and 
also  by  compression.     Heat  due  to  compression   is  appreciable 


Heat  and  Temperature  163 

only  in  gases,  and  is  familiar  in  the  heating  of  a  bicycle  pump 
when  in  use  {Exp.).  The  heating  is  not  due  to  the  friction  of 
the  piston,  to  any  appreciable  extent,  but  to  the  compression  of 
the  air.  The  heat  thus  developed  is,  in  fact,  the  exact  equivalent 
of  the  mechanical  work  done  in  compressing  the  air. 

Chemical  action  is  the  most  important  source  of  heat,  with  the 
exception  of  the  sun.  The  burning  of  any  substance  is  an  exam- 
ple. When  a  substance  burns,  its  molecules  are  broken  up,  and 
their  parts  (atoms)  unite  with  oxygen  from  the  air.  New  sub- 
stances are  thus  formed,  which  are  in  most  cases  gases.  It  is  not 
surprising  that,  in  this  violent  rearrangement  of  molecular  struc- 
ture, the  motion  of  the  molecules  should  be  greatly  increased. 
Heat  is  also  generated  by  other  chemical  processes.  For  exam- 
ple, when  water  is  poured  on  quicklime,  the  two  unite  chemically, 
forming  slaked  lime,  and  the  lime  becomes  very  hot  {Exp.). 
Similarly,  when  strong  sulphuric  acid  is  poured  into  water,  the 
mixture  becomes  very  hot  {Exp.). 

The  transformation  of  other  forms  of  energy  into  heat  is  con- 
sidered in  connection  with  other  topics. 

213.  Temperature.  —  The  relation  between  heat  and  tempera- 
ture is  similar  to  that  between  quantity  of  water  and  water  level. 
If  two  vessels  containing  water  are  in  communication,  we  know 
that  the  flow  will  be  from  the  one  in  which  the  water  stands  at 
the  higher  level  to  the  other,  whether  the  one  or  the  other  con- 
tains the  greater  quantity  of  water.  Similarly,  the  transfer  of  heat 
between  two  bodies  or  between  parts  of  the  same  body  is  always 
from  the  hotter  to  the  cooler,  whether  the  quantity  of  heat  in  the 
former  be  greater  or  less  than  that  in  the  latter ;  and  this  transfer- 
ence will  cease  as  soon  as  both  are  at  the  same  temperature. 

Temperature  is  therefore  defined  as  the  condition  of  bodies  that 
determines  the  direction  in  which  the  transfer  of  heat  can  take 
place  between  them.  Or,  since  the  temperature  of  a  body  is 
higher  the  greater  the  amount  of  heat  it  contains,  temperature 
may  be  defined  as  the  ititeiisity  or  degree  of  heat  (not  the  quantity 
of  heat). 


164  Heat 

214.  Temperature  Sensations.  —  Our  bodily  sensations  of  heat 
and  cold  afford  direct  but  inexact  and  often  misleading  infor- 
mation concerning  the  temperature  of  bodies ;  they  afford  no 
information  whatever  concerning  the  relative  amount  of  heat  in 
different  bodies  (Art.  217). 

PROBLEMS 

1.  What  is  ihe  meaning  of  the  adjective  cold?  of  the  noun  cold? 

2.  A  body  {x>sscsses  heat  as  lung  as  it  is  capable  of  becoming  colder.  Is 
there  any  heat  in  ice? 

3.  (a)  According  to  the  theory  of  heat,  what  would  be  the  molecular 
condition  of  a  body  having  no  heat?  {h)  Why  could  not  such  a  body  be  a  gas? 

4.  Mention  any  familiar  instances  in  which  ccjual  temperatures  do  not 
cause  equal  temiierature  sensations. 

II.  The  Transmission  of  Heat 

215.  The  transference  of  heat  as  heat  from  one  place  to 
another  takes  place  in  two  ways;  namely,  by  conduction  and  by 
convection. 

Laboratory  Exercise  .?/. 

216.  Conduction.  —  Conduction  is  the  transmission  of  heat  from 
hotter  to  colder  parts  of  a  body,  or  from  a  hotter  to  a  colder  body  in 
contact  with  it,  without  change  in  the  relative  positions  of  the  parts 
of  the  body.  It  is  the  only  process  by  which  heat  travels  in  solids. 
The  heating  of  the  farther  end  of  a  poker  when  one  end  is  placed 
in  a  fire,  and  the  heating  of  the  handle  of  a  spoon  placed  in  a  cup 
of  hot  tea  are  familiar  examples. 

The  kinetic  theory  suggests  a  mental  picture  of  the  process  of 
heat  conduction.  When  any  part  of  a  body  is  heated,  its  mole- 
cules are  set  in  more  rapid  vibration.  These  molecules  jostle 
their  neighbors  more  violently,  increasing  the  energy  of  their  vibra- 
tion. The  disturbance  thus  spreads  throughout  the  body  without 
change  in  the  relative  positions  of  the  molecules  themselves.  In 
conduction,  therefore,  molecular  energy  is  transmitted  without  the 
transmission  of  matter. 

Substances  differ  widely  in  their  conductivity,  i.e.  their  power  of 


The  Transmission  of  Heat 


165 


transmitting  heat  by  conduction.  The  metals  are  the  best  con- 
ductors of  heat ;  other  solids,  with 
few  exceptions,  are  better  conduct- 
ors than  liquids.  Liquids,  with 
the  exception  of  mercury  and  mol- 
ten metals,  are  very  poor  conduct- 
ors. Water  can  be  boiled  at  the 
top  of  a  test  tube  for  several  min- 
utes, while  the  greater  part  of  it 
remains  cold  (Fig.  119)  (£x/>.). 
But  wood  and  paper,  especially  the 
latter,  are  much  poorer  conductors 
than  water  (see  table  below).  Gases  are  practically  nonconduct- 
ors. In  testing  the  conductivity  of  liquids  and  gases  they  must 
be  heated  at  the  top  to  prevent  currents  (Art.  218). 

The  following  table  gives  the  conductivities  of  various  substances 
referred  to  silver  as  the  standard  :  — 


Fig.  119. 


Tad/e  of  Conductivities  for  Heat.     (^Approximations^ 


Silver icxj 

Copper 74 

Brass 27 

Iron 12 

Lead 8.5 

Mercury 1.35 

Ice 0.20 


Glass 0.20 

Marble 0.15 

Water 0.14  . 

Wood 0.04 

Writing  paper 0.012 

Wool 0.009 

Air 0.005 


217.  Illustrations  of  Good  and  Poor  Conductivity.  —  On  a  cold 
morning  the  ^oox  feels  colder  to  the  bare  feet  than  the  carpet  or  a 
rug,  and  water  that  has  been  standing  in  the  room  over  night  feels 
colder  to  the  face  than  the  air.  These  and  other  objects  in  the 
room,  which  seem  to  be  unequally  cold,  are  all  at  the  same  tem- 
perature as  the  air.  The  difference  in  the  sensations  is  largely  due 
to  the  difference  in  the  conductivities  of  the  substances  touched. 
If  the  body  touched  is  a  good  conductor,  heat  is  rapidly  conducted 
to  all  parts  of  it  from  the  hand  \  and  this  continue^  a§  long  as  the 


1 66  Heat 

hand  remains  in  contact  with  it  or  until  it  becomes  warm  through- 
out. This  rapid  and  continued  loss  of  heat  makes  the  hand  cold. 
If  the  body  touched  is  a  poor  conductor,  but  little  heat  is  lost  from 
the  hand  in  warming  the  part  of  it  that  is  touched.  Similarly,  if 
hot  substances  at  the  same  temperature  but  of  different  conduc- 
tivities are  touched,  the  best  conductors  feel  the  hottest  because 
they  conduct  heat  to  the  hand  most  rapidly,  and  hence  make  the 
hand  hotter  than  the  poorer  conductors  do.  (The  temperature 
sensation  caused  by  a  body  depends  also  in  part  upon  another 
property,  called  specific  heat,  which  will  be  studied  later.) 

The  low  conducting  power  of  air  is  utilized  in  refrigerators  and 
ice  houses,  which  have  double  walls  filled  between  with  charcoal, 
sawdust,  straw,  or  other  loose,  badly  conducting  material,  which 
hinders  the  circulation  of  the  air.  The  warmth  of  fur,  feathers,  and 
wool  is  partly  due  to  the  air  entangled  in  them. 

PROBLEMS 

1.  What  is  the  advantage  of  the  poor  conductivity  of  wood  in  using 
matches? 

2.  An  overcoat  is  said  to  "  keep  out  the  cold."  What  is  it  that  it  really 
does? 

3.  Why  is  woolen  clothing  warmer  than  cotton  or  linen? 

4.  Would  it  l>e  better  to  wrap  a  piece  of  ice  in  a  woolen  or  a  cotton 
cloth  to  keep  it  from  melting? 

5.  What  is  the  advantage  of  having  the  bottoms  of  tin  teakettles  and 
boilers  made  of  copper? 

Laboratory  Exercise  28. 

218.  Convection. — The  transmission  of  heat  in  a  liquid  or  a 
gas  by  currents  due  to  unequal  temperatures  in  its  different  parts 
is  called  convection ,  and  the  currents  are  called  convection  cur- 
rents. 

In  heating  liquids,  the  heat  is  applied  to  the  lx)ttom  of  the 
vessel.  The  liquid  at  the  bottom  receives  heat  from  the  vessel  by 
conduction  and  expands,  thus  becoming  less  dense.  It  is  then 
displaced  by  the  denser  liquid  at  the  top,  producing  convection 
currents. 


Radiation  167 

Convection  currents  in  air  and  other  gases  are  due  to  the  same 
cause.  The  strong  ascending  current  above  a  bonfire  is  indicated 
by  the  leaping  of  the  flames  and  the  rapid  rise  of  sparks  and 
smoke.  The  fire  is  fed  by  inward-flowing  currents  near  the 
ground.  They  occupy  much  more  space  than  the  ascending 
current,  and  hence  move  more  slowly  and  are  less  noticeable. 

PROBLEMS 

1.  Would  convection  currents  be  caused  by  heating  a  liquid  at  the  top? 
by  cooling  it  at  the  top?    l)y  cooling  it  at  the  bottom? 

2.  What  would  be  the  general  direction  of  convection  currents  in  a  room 
heated  by  a  stove  at  one  end  of  the  room?  Would  the  air  be  warmer  near 
the  floor  or  near  the  ceiling? 

3.  What  convection  currents  are  set  up  when  a  door  is  left  open  between  a 
warm  and  a  cold  room  ? 

4.  Why  do  openings  at  the  top  and  bottom  of  a  window  provide  better 
ventilation  than  a  single  opening  of  the  same  total  area  at  either  top  or 
bottom? 

5.  A  fire  in  a  fireplace  provides  excellent  ventilation  for  a  room.     Explain. 

6.  Docs  an  open  fireplace  provide  as  good  ventilation  whether  there  is  a 
fire  in  it  or  not? 

7.  How  is  the  flame  of  a  lamp  provided  with  the  constant  supply  of 
oxygen  necessary  for  combustion? 

III.   Radiation 

Laboratory  Exercise  2^  {l^  a  and  <5,  and  II,  dr  and  b), 
219.  An  Experiment  on  Radiation.  —  When  the  hand  is  held 
close  besiWe  a  hot  flame,  as  that  of  a  Bunsen  burner,  the  side  of  the 
hand  turned  toward  the  flame  becomes  hot.  The  hand  is  not 
heated  by  convection,  for  it  is  in  the  path  of  the  currents  of  cold 
air  moving  toward  the  flame.  When  any  object,  as  a  sheet  of 
paper,  is  placed  as  a  screen  between  the  hand  and  the  flame,  the 
hand  instantly  ceases  to  feel  the  heat,  which  shows  that  it  could 
not  have  been  heated  by  conduction,  for  three  reasons:  (i)  be- 
cause the  screen  is  at  least  as  good  a  conductor  as  the  air ;  (2)  if 
the  process  were  conduction,  the  air  must  have  been  at  least  as 
warm  as  the  hand  and  it  could  not  instantly  become  cold  on  inter- 


1 68  Heat 

posing  the  screen ;  and  (3)  both  sides  of  the  hand  would  have 
been  warmed,  for  convection  currents  would  prevent  any  consider- 
able difference  in  the  temperature  of  the  air  on  the  two  sides. 
The  conclusion  is  therefore  certain  that,  when  the  hand  is  beside 
the  flame  with  nothing  interposed  between,  the  side  turned  toward 
the  flame  becomes  much  hotter  than  the  air  at  that  distance,  and 
that  it  is  heated  neither  by  conduction  nor  convection. 

The  conditions  of  this  experiment  are  repeated  on  a  large  scale 
when  we  stand  near  a  bonfire  in  cold  weather.  Most  people  know 
by  experience  of  the  last  situation  what  it  means  to  "  roast  on  one 
side  and  freeze  on  the  other." 

220.  The  Transformation  of  Heat  into  Radiant  Energy  and  of 
Radiant  Energy  into  Heat.  —  All  hot  bodies  lose  heat  by  a  pro- 
cess called  ratiiation,  which  is  independent  of  conduction  and  con- 
vection and  is  totally  different  from  either.  But  radiation  is  not  a 
process  by  which  heat  is  transmitted ;  for  the  heat  is  transformed 
at  the  radiating  body  into  another  form  of  energy,  called  radiant 
energy^  and  is  transmitted  as  such  until  it  meets  some  body  capa- 
ble of  transforming  it  back  into  heat. 

Tlie  transformation  of  radiant  energy  into  heat  is  called  absorp- 
tion. Absorption  takes  place  at  the  surface  of  some  bodies,  as  at 
the  surface  of  the  hand  in  the  preceding  experiment ;  in  other 
cases  it  takes  place  within  the  body  through  which  the  radiant 
energy  is  passing.  A  body  is  heated  only  by  the  part  of  the  radiant 
energy  that  it  absorbs.  Air  absorbs  comparatively  little,  hence  is 
only  slightly  heated  by  radiation.  The  transformation  of  radiant 
energy  into  heat  by  absorption  may  l>e  compared  to  the  transfor- 
mation of  the  kinetic  energy  of  a  flying  bullet  into  heat  by  impact 
against  a  steel  target.  Radiant  energy  is  as  distinctly  dififerent 
from  heat  as  is  the  kinetic  energy  of  the  bullet. 

221.  Luminous  and  Nonluminous  Radiation.  —  Radiant  energy 
is  often  called  radiation ;  thus  radiation  may  mean  either  a  form 
of  energy  or  the  process  by  which  that  energy  is  transmitted. 
Radiant  energy  includes  luminous  radiation,  or  lights  as  well  as  the 
Donluminous  or  dark  radiation  previously  considered.     The  dis- 


Radiation  169 

tinction  between  light  and  nonluminous  radiation  is  a  physiologi- 
cal rather  than  a  physical  one,  and  is  due  to  the  fact  that  the 
optic  nerve  is  sensitive  to  the  one  and  not  to  the  other.  The  two 
kinds  of  radiation  are  the  same  form  of  energy. 

A  hot  body,  as  a  piece  of  iron,  gives  out  only  dark  radiation 
below  a  certain  temperature ;  at  "  red  heat "  or  above  it  radiates 
light  also.  Light  is  due  to  the  transformation  of  heat  into  radiant 
energy  at  sufficiently  high  temperatures ;  and,  like  dark  radiation, 
it  is  again  transformed  into  heat  by  absorption. 

The  energy  of  dark  radiation  is  greater  than  that  of  light ;  hence 
it  causes  greater  heating  when  absorbed.  Before  heat  was  recog- 
nized as  a  form  of  energy  (Art.  210)  it  was  supposed  that  dark 
radiation  was  a  form  of  heat  and  was  essentially  different  from 
light ;  it  was  therefore  called  "  radiant  heat,"  and  its  transmission 
was  called  the  "  radiation  of  heat."  These  terms  are  still  com- 
mon ;  but  their  use  should  be  avoided,  at  least  by  the  beginner. 

222.  Radiant  Energy  can  be  transmitted  in  a  Vacuum. — 
Neither  air  nor  any  other  form  of  matter  that  we  have  yet  con- 
sidered is  necessary  for  the  transmission  of  radiation.  This  is 
evident  from  the  fact  that  we  receive  radiant  energy  from  the  sun 
over  a  distance  of  93,000,000  miles ;  through  all  of  which  space, 
till  the  earth's  atmosphere  is  reached,  there  is  what  we  call  a 
perfect  vacuum.  The  incandescent  electric  light  affords  another 
illustration  on  a  small  scale.  When  the  filament  of  the  lamp  is 
heated  by  the  electric  current,  it  sends  out  both  luminous  and 
nonluminous  radiation,  yet  within  the  bulb  there  is  a  very  nearly 
perfect  vacuum. 

Hitherto  we  have  found  energy  only  in  connection  with  matter  ; 
in  fact,  matter  is  often  called  "  the  vehicle  of  energy."  How, 
then,  does  energy  travel  in  a  vacuum  ?  The  question  is  answered 
by  the  theory  of  radiant  energy ;  which,  together  with  the  laws 
of  radiation,  will  be  considered  under  the  subject  of  light.  One 
law  of  radiation  was  illustrated  when  the  screen  was  interposed 
between  the  Bunsen  flame  and  the  hand  :  the  radiation  did  not 
pass  round  the  screen.     Radiation  is  transmitted  along  a  straight 


170 


Heat 


path  in  a  vacuum  and  in  all  homogeneous  substances  capable  of 
transmitting  it. 

223.  The  Radiometer.  —  The  radiometer  (Fig.  1 20)  consists  of 
four  light  vanes  of  mica  or  aluminum  attached  to  a  vertical  axis, 
and  inclosed  in  a  glass  bulb  containing  a  small  quantity  of  air 

under  very  low  pressure.  One  side  of  each 
vane  is  bright,  the  other  is  coated  with 
lampblack.  When  the  instrument  is 
placed  in  the  sunshine  or  in  the  path  of 
other  radiation,  the  vanes  rotate  with  their 
bright  side  in  advance. 

The  rotation  of  the  vanes  is  explained 
as  follows  :  The  black  surfaces  absorb 
more  radiation  than  the  bright,  and  hence 
are  warmer.  The  molecules  of  the  rare- 
fied air  that  strike  the  blackened  surfaces 
are  heated,  and  rebound  with  greater  veloc- 
ity than  those  that  strike  the  bright  sides. 
This  causes  a  greater  pressure  upon  the 
black  side  of  each  vane  than  upon  the 
other  ;  hence  the  rotation.  If  the  air  in  the  bulb  were  not 
highly  rarefied,  there  would  be  no  rotation,  for  the  collisions 
among  the  molecules  would  be  so  frequent  as  to  equalize  the  pres- 
sures throughout  the  bulb. 

The  rate  of  rotation  of  the  vanes  serves  as  a  rough  measure  of 
the  energy  of  the  radiation  falling  upon  them ;  hence  the  instru- 
ment is  very  useful  in  the  study  of  radiation  (see  Lab.  Ex.  29). 

224.  Effects  of  Matter  upon  Radiation.  —  When  radiant  energy 
falls  upon  any  substance,  it  may  be  (i)  absorbed  at  the  surface, 
(2)  reflected  by  the  surface,  (3)  transmitted  through  the  sub- 
stance, or  (4)  wholly  or  partly  absorbed  by  the  substance  during 
transmission.  Generally  two  and  often  three  of  these  effects 
occur  simultaneously  with  different  portions  of  the  radiation.  The 
study  of  these  effects  will  be  continued  under  the  subject  of  light. 

225.  Reflection  and  Absorption  at  Surfaces.  —  In  general,  the 


Fig.  laa 


Radiation  171 

surfaces  of  substances  that  transmit  no  radiation  reflect  part  and 
absorb  the  remainder ;  they  are  heated  only  by  the  absorbed 
radiation.  Lampblack  is  the  best  absorber  known ;  it  absorbs 
practically  all  of  the  radiation  that  falls  upon  it,  both  luminous  and 
nonluminous.  Any  polished  metal  reflects  much  the  greater  part 
of  all  radiation  and  absorbs  the  remainder.  A  piece  of  tin  coated 
with  lampblack  or  painted  black  on  the  side  turned  toward  a  flame 
will  therefore  become  hot,  while  a  piece  of  bright  tin  in  the  same 
situation  will  be  only  slightly  warmed.  A  white  surface  reflects 
nearly  all  luminous  radiation  that  falls  upon  it,  a  black  surface  re- 
flects almost  none,  a  colored  surface  reflects  part  and  absorbs  part. 

226.  Selective  Absorption.  —  Substances  that  transmit  d^irk 
radiation,  with  but  little  absorption  if  any,  are  called  diatherma- 
nous  (Greek  therme,  heat)  ;  those  that  transmit  little  or  none  are 
called  athennanous.  The  terms  have  the  same  meaning  with 
respect  to  dark  radiation  that  transparent  and  opaque  have  with 
respect  to  light. 

The  power  of  most  substances  to  transmit  radiation  depends 
very  largely  upon  the  temperature  of  the  source  of  the  radiation. 
Some  substances  that  transmit  light  also  transinit  dark  radiation,^ 
others  do  not.  Clear  glass  transmits  light  and  also  a  considerable 
portion  of  the  radiation  from  bodies  nearly  red  hot,  but  absorbs 
all  or  nearly  all  of  the  radiation  from  colder  bodies.  Thus  solar 
radiation  enters  a  sunny  room  in  large  quantities  through  the 
windows,  and  is  absorbed  by  the  objects  upon  which  it  falls.  But, 
as  the  radiation  from  these  bodies  cannot  penetrate  glass,  the  solar 
energy  is  entrapped  in  the  room,  which  may  thus  become  con- 
siderably warmer  than  the  air  outside.  This  explains  the  accu- 
mulation of  heat  in  greenhouses.  Rock  salt  transmits  all  radiation, 
being  highly  transparent  and  diathermanous.  A  solution  of  iodine 
in  carbon  disulphide  is  perfectly  opaque  but  diathermanous ; 
water  in  its  three  states  is  transparent  but  highly  athermanous 
(Lab.  Ex.  29). 

The  unequal  absorption  of  luminous  and  dark  radiation  by  the 
same  substance  is  called  selective  absorption. 


172  Heat 

227.  Relation  of  Radiating  and  Absorbing  Powers.  —  If  two 
metal  vessels  of  the  same  size  and  material,  one  highly  polished 
and  the  other  coated  on  the  outside  with  lampblack,  are  filled 
with  equal  quantities  of  hot  water  at  the  same  temperature  and  are 
allowed  to  stand  for  some  minutes,  it  will  be  found  that  the  tem- 
perature of  the  water  in  the  blackened  vessel  is  considerably  lower 
than  that  in  the  other  {Exp.).  The  difference  in  the  rate  of  cool- 
ing is  due  to  the  more  rapid  radiation  from  the  blackened  surface. 
The  experiment  illustrates  the  fact  that  good  absorbers  of  radiation 
are  also  good  radiators,  and  poor  absorbers  poor  radiators. 

228.  The  Heating  of  the  Atmosphere.  —  It  is  well  known  that 
the  atmosphere  is  colder  at  higher  than  at  lower  altitudes.  Aero- 
nauts always  find  intense  cold  at  altitudes  above  three  or  four 
miles,  and  the  tops  of  high  mountains  are  covered  with  perpetual 
snow.  The  low  temperature  of  high  altitudes  is  due  to  the  fact 
that  the  dry  and  rarefied  air  absorbs  but  little  of  the  enormous 
quantity  of  radiant  energy  that  passes  through  it. 

As  the  radiation  approaches  the  earth,  the  rate  of  absorption 
rapidly  increases,  principally  on  account  of  the  greater  amount  of 
water  vapor  ;  for  experiments  have  shown  that  the  absorbing 
power  of  air  containing  the  average  amount  of  water  vapor  is 
seventy-two  times  as  great  as  that  of  perfectly  dry  air  of  the  same 
density.  The  absorption  at  lower  levels  is  further  increased  by  the 
dust  particles  in  the  air. 

But  notwithstanding  the  loss  by  absorption  on  the  way  through 
the  atmosphere,  it  is  estimated  that  from  two  thirds  to  three  fourths 
of  the  solar  radiation  reaches  the  earth.  Much  of  this  is  absorbed 
by  the  surface  of  the  land  and  the  ocean ;  the  remainder  is  re- 
flected. The  reflected  radiation  is  partly  absorbed  on  its  way  out 
through  the  atmosphere  again.  The  absorbed  radiation  warms  the 
surface  of  the  land,  and  this  in  turn  warms  the  air  in  contact  with 
it.  This  is  especially  noticeable  on  a  hot  day  in  summer,  when, 
if  there  is  no  wind,  the  air  close  to  the  ground  is  many  degrees 
warmer  than  at  a  height  of  a  few  feet.  The  heating  of  the  air  at 
the  bottom  causes  convection  currents  (winds),  by  which  the  heat 


Temperature  and  Expansion  173 

is  carried  to  considerable  altitudes;  but  the  temperature  is,  of 
course,  highest  at  the  source  of  the  heat^  i.e.  at  the  earth's  sur- 
face. 

The  earth  is  cooled  at  night  principally  by  radiation.  The  loss 
is  rapid  on  clear  nights,  especially  when  the  atmosphere  is  very 
dry ;  but  is  checked  in  a  large  degree  by  moisture,  especially  by 
clouds,  which  absorb  the  radiation  before  it  has  passed  through 
them,  thus  serving  as  a  blanket  to  the  earth.  Hence  clear  nights 
are,  as  a  rule,  the  coldest  At  high  altitudes,  where  there  is  but 
little  hindrance  to  radiation  either  by  day  or  by  night,  sheltered 
valleys  are  quickly  warmed  in  summer  by  the  early  morning  sun- 
shine ;  and  a  sudden  chill  immediately  follows  the  disappearance 
of  the  sun  in  the  evening,  the  nights  being  often  cold  enough  for 
frost. 

Thus  we  see  that  the  atmosphere,  or  rather  the  moisture  in  it, 
performs  an  indispensable  function  in  moderating  the  intensity  of 
solar  radiation  by  day  and  retaining  the  heat  by  night. 

PROBLEMS 

1.  Why  does  snow  melt  more  quickly  when  covered  with  a  thin  layer  of 
earth? 

2.  Why  is  Hght-colored  clothing  more  comfortable  in  summer  than  black? 

3.  Why  is  the  difference  between  the  temperature  in  the  sunlight  and  in 
the  shade  greater  upon  the  top  of  a  mountain  than  at  a  low  elevation? 

4.  Why  must  those  who  climb  snow-covered  mountains  take  especial  care 
to  protect  their  faces?' 


IV.    Measurement  of  Temperature  and  Expansion 

229.  Measurement  of  Temperature.  —  Any  instrument  for  meas- 
uring temperatures  is  called  a  thermometer.  In  most  forms  of 
thermometers  the  effect  of  heat  in  changing  the  volume  of  some 
substance  is  utilized.  Solids  are  rarely  used,  as  their  expansion  is 
small  and  they  are  otherwise  inconvenient.  It  is  important  that 
the  substance  chosen  should  have  a  uniform  expansion  ;  i.e.  equal 


174  Heat 


quantities  of  heat  should,  cause  equal  increases  of  volume  at  all 
temperatures. 

Mercury  fulfills  this  condition  better  than  any  other  liquid,  and 
has  the  further  advantage  of  remaining  a  liquid  through  a  very 
wide  range  of  temperature.  The  apparent  expansion  of  mercury 
in  a  glass  vessel  (/.<•.  the  difference  between  the  expansion  of 
mercury  and  glass)  has  therefore  been  adopted  as  the  standard. 
For  temperatures  below  the  freezing  point  of  mercury,  alcohol 
thermometers  are  used,  the  freezing  point  of  alcohol  being  —  i3o°C. 
The  air  thermometer,  in  which  the  expansion  of  air  is  utilized,  is 
adopted  as  the  standard  in  the  most  accurate  scientific  work  ;  and 
it  can  be  constructed  of  such  form  as  to  be  of  practical  use  in  meas- 
uring temperatures  above  the  boiling  point  of  mercury,  such  as 
the  temperatures  of  furnaces. 

230.  The  Mercury  Thermometer.  —  The  mercury  thermometer 
consists  essentially  of  a  capillary  glass  tube,  called  the  stem,  ter- 
minating in  a  bulb  (Fig.  122).  The  bulb  and  a  part  of  the  stem 
are  filled  with  mercury,  and  the  expansion  is  measured  by  a  scale 
engraved  upon  the  stem  or  attached  to  it.  In  making  a  thermom- 
eter the  mercury  is  heated  to  drive  out  all  the  air  before  the  stem 
is  sealed  at  the  top ;  hence  the  space  in  the  tube  above  the  mer- 
cury is  a  vacuum. 

231.  Determination  of  the  Fixed  Points. — Probably  no  two 
thermometers  have  bulbs  of  exactly  the  same  capacity  and  tubes 
of  exactly  the  same  bore ;  hence  the  readings  of  different  ther- 
mometers would  be  entirely  inconsistent  with  one  another  if  they 
were  provided  with  scales  of  equal  length.  The  correct  position 
and  dimensions  of  the  scale  must  therefore  be  determined  sepa- 
rately for  every  thermometer. 

The  first  step  in  this  process  is  to  determine  the  "  fixed  points," 
called  \k\^  freezing  point  diViA  the  boiling  point.  T\\^  freezing  point 
is  the  temperature  at  which  pure  water  freezes ;  but  since  this  is 
exactly  the  same  as  the  temperature  at  which  ice  melts,  whatever 
the  surrounding  temperature  may  be,  it  is  most  conveniently 
found  by  inserting  the  bulb  of  the  thermometer  in  a  dish  of  melt- 


Temperature  and   Expansion 


^75 


ing  snow  or  crushed  ice.  The  ice  is  packed  about  the  bulb  and 
stem,  leaving  the  mercury  just  visible  above  it ;  and  a  mark  is  made 
on  the  stem  at  the  top  of  the  mercury  column  after  it  comes  to  rest. 

The  boiling  point  is  the  temperature  at  which  pure  water  boils 
under  a  pressure  of  one  atmos- 
phere. But,  since  the  tempera- 
ture of  the  water  is  subject  to 
appreciable  changes  from  various 
causes  while  that  of  the  escaping 
steam  is  constant,  the  thermom- 
eter is  adjusted  so  as  to  be  sur- 
rounded by  the  steam,  as  nearly  as 
possible  to  the  top  of  the  mercury 
in  the  stem,  and  is  not  permitted 
to  touch  the  water  (Fig.  121). 
Since  even  a  slight  change  in  the 
atmospheric  pressure  causes  an 
appreciable  change  in  the  tem- 
perature at  which  water  boils 
(Art.  262),  a  correction  must  be  applied  to  the  observed  height 
of  the  mercury  in  the  stem  if  tlie  barometric  pressure  is  not 
76  cm.  when  the  boihng  point  is  determined. 

232.  Centigrade  and  Fahrenheit  Scales.  —  The  distance  between 
the  fixed  points  is  divided  into  equal  parts  called  degrees.  In  the 
Centigrade  scale  the  number  of  these  divisions  is  100,  the  freezing 
point  being  marked  0°  and  the  boiling  point  100°.  The  Centi- 
grade thermometer  is  almost  exclusively  used  in  scientific  work. 
All  temperatures  referred  to  in  this  book  are  expressed  in  the  Cen- 
tigrade scale  unless  otherwise  indicated.  In  the  Fahrenheit-  scale 
the  freezing  point  is  marked  32°  and  the  boiling  point  212°,  the 
interval  between  them  being  180°.  The  Fahrenheit  scale  is  the 
one  in  general  use  in  this  country.  The  scale  of  a  thermometer 
may  be  extended  to  any  desired  distance  beyond  the  fixed  points. 
Temperatures  below  zero  on  either  scale  are  indicated  by  the  nega- 
tive sign,  as  —  15°  C. 


Fig.  lai. 


or  THr 


176 


Heat 


c 

00 

F 

212 

0 

3? 

■a 

FlO.    122. 


Since  the  interval  between  the  fixed  points  is  loo  Centigrade 
degrees  or  i8o  Fahrenheit  degrees,  it  follows 
that  — 

I   Centigrade  degree  =  f  Fahrenheit  degrees, 
and 

I  Fahrenheit  degree  =  f  Centigrade  degrees. 
In  changing  a  reading  on  either  scale  to  the 
equivalent  reading  on  the  other,  allowance  must 
be  made  for  the  difference  in  the  zero  points. 
Example :  50°  C.  means  50  Centigrade  degrees 
above  the  freezing  point.  This  is  equal  to  50  x  |, 
or  90  Fahrenheit  degrees  a  dove  the  freezing  pointy 
or  to  122°  F. 

233.  Linear  Expansion  of  Solids.  —  With  few 
exceptions,  none  of  which  are  important,  solids  expand  when 
heated  and  contract  when  cooled.  Tiie  total  expansion  or  con- 
traction of  any  body  depends  upon  its  size  and  material  as  well 
as  upon  the  change  in  temperature.  The  expansion  of  a  solid 
takes  place,  of  course,  in  its  three  dimensions ;  but,  in  most  cases, 
it  is  important  only  in  the  direction  of  its  length.  F^xpansion, 
when  considered  in  one  direction  only,  is  called  linear  expansion. 
The  expansion  of  a  solid  per  unit  of  length  when  its  tempera- 
ture rises  one  degree  is  called  its  coefficient  of  linear  expansion. 
For  example,  the  coefficient  of  linear  expansion  of  oak  is  .000006, 
which  means  that  each  centimeter  of  its  length  increases  to 
1.000006  cm.  with  a  rise  of  temperature  of  one  degree  Centi- 
grade. The  coefficient  of  steel  is  .000012 — just  twice  that  of 
oak  ;  hence  the  expansion  of  a  piece  of  steel  for  any  change 
of  temperature  is  twice  as  great  as  that  of  a  piece  of  oak  of  the 
same  dimensions  for  the  same  change  of  temperature. 

The  expansion  of  solids  is  so  slight  that  some  special  device 
must  be  employed  in  order  to  measure  with  any  degree  of  accu- 
racy the  expansion  of  even  a  long  rod  for  a  considerable  rise  of 
temperature.  The  methods  by  which  this  is  done  are  best  studied 
in  the  laboratory.     The  computation  of  the  coefficient  from  the 


Temperature  and  Expansion  177 

data  thus  obtained  is  illustrated  by  the  following  example  :  A  brass 
rod  90  cm.  long  expands  .13  cm.  when  heated  from  23°  to  100°. 
Find  its  coefficient  of  linear  expansion.  The  rise  of  temperature 
is  77  degrees;  hence,  assuming  the  expansion  to  be  uniform^  the 

expansion  of  the  rod  for  a  rise  of  temperature  of  i  degree  is  — 
cm.,  and  the  expansion  of  i  cm.  for  a  rise  of  temperature  of  i  de- 
gree is  — '—^ — ,  or  .0000188  cm. 
^  77  X  90' 

The  coefficient  of  linear  expansion  may  equally  well  be  re- 
garded as  the  ratio  of  the  whole  expansion  to  the  whole  length  for 
a  rise  of  temperature  of  one  degree. 


Coefficients  of  Linear  Expansion 


Zinc o.(xxx>294 

Lead 0.0000286 

Aluminum 0.000023 

Brass 0.0000188 

Copper      ......  0.0000172 


Iron  and  Steel  ....  o.ooooi  22 

Platinum 0.0000088 

Glass 0.0000086 

Wood,  Dak 0.000006 

Wood,  Fir 0.0000035 


Laboratory  Exercise  JO. 

234.  Effects  and  Applications  of  Expansion. — The  force  that 
a  body  can  exert  in  expanding  or  contracting  with  change  of 
temperature  is  equal  to  the  force  required  to  expand  or  compress 
it  to  the  same  extent  by  mechanical  means.  This  force  is  enor- 
mous, and,  under  most  circumstances,  practically  irresistible.  If 
a  bar  of  malleable  iron  one  square  inch  in  cross-section  were 
placed  between  fixed  supports,  so  as  to  make  expansion  impossible, 
and  its  temperature  then  raised  40°,  it  would  exert  a  pressure  of 
about  five  tons  against  the  supports. 

The  rails  of  tracks  are  laid  with  a  small  space  between  their 
ends,  which  provides  room  for  expansion.  A  long  steel  bridge 
changes  in  length  several  inches  between  winter  and  summer, 
opportunity  for  this  change  being  afforded  by  an  expansion  joint. 
The  tire  of  a  wagon  wheel  is  made  just  large  enough    to  go  on 


178  Heat 

when  hot ;  it  shrinks  upon  the  wheel  in  cooling,  making  a  very 
tight  fit.  For  a  similar  reason  red-hot  rivets  are  used  in  joining 
the  steel  plates  of  tanks  and  boilers.  A  wooden  rod  is  better  than 
one  of  metal  for  the  pendulum  of  a  clock,  since  its  coefficient  of 
expansion  is  less. 

235.  The  Expansion  of  Liquids.  —  In  considering  the  expan- 
sion of  liquids  and  gases  it  is  increase  of  volume,  or  cubical  expan- 
sion^ with  which  we  are  concerned.  The  coefficient  of  cubical 
expansion  of  a  solid  or  a  liquid  is  its  expansion  per  unit  volume 
for  a  rise  of  temperature  of  one  degree. 

As  usually  contrived,  experiments  on  the  expansion  of  liquids 
give  their  apparent  expansion,  i.e,  the  difference  between  their 
true  expansion  and  the  expansion  of  the  containing  vessel.  The 
tnie  expansion  of  a  liquid  is  the  sum  of  its  apparent  expansion 
and  the  cubical  expansion  of  the  material  of  the  containing  vessel. 
In  general,  a  liquid  expands  more  rapidly  as  the  temperature 
approaches  its  boiling  point.  The  following  table  gives  the  true 
average  expansion  of  a  few  liquids  :  — 

Coefficients  of  Cubical  Expansion 


Ether 0.0015 

Alcohol  (5°  to  6**)  .     .     .     .  0.00105 
Alcohol  (49°  to  50**)  .    .     .  0.00122 

Acetic  acid 0.00105 

Petroleum 0.0009 

Olive  oil 0.0008 


Turpentine 0.0007 

Glycerine 0.0005 

Water  (5°  to  6°)  .     .     .     .  0.000022 
Water  (49°  to  50°)    .     .     .  0,00046 
Water  (99°  to  100")  .     .     .  0.00076 
Mercury 0.00018 


It  can  be  shown  that  the  coefficient  of  cubical  expansion  of  a 
solid  is  three  times  its  coefficient  of  linear  expansion ;  hence  by 
multiplying  the  values  given  in  the  table  under  solids  by  three,  a 
comparison  can  be  made  between  the  expansion  of  liquids  and 
solids.  It  will  be  found  that  the  expansion  of  liquids,  though 
small,  is  considerably  greater  than  that  of  solids.  The  total 
expansion  of  water  between  4°  and  100°  is  a  little  over  four  per 
cent. 

Laboratory  Exercise  ji. 


Temperature  and  Expansion  179 

236.  Importance  of  the  Irregular  Expansion  of  Water.  —  The 
expansion  of  water  is  curiously  irregular.  It  contracts  as  its  tem- 
perature rises  from  0°  to  4° ;  when  heated  beyond  this  point  it 
begins  to  expand,  at  first  very  slowly,  then  more  and  more  rapidly 
(see  table).  Hence  the  density  of  water  is  greatest  at  4°  C. 
(about  39°  F.). 

This  behavior  of  water  is  of  the  greatest  importance  in  the 
economy  of  nature.  In  winter  the  waters  of  lakes  lose  heat  at 
the  surface  by  contact  with  the  cold  air  and  by  radiation.  As  the 
water  at  the  surface  cools,  it  becomes  denser  and  sinks,  displacing 
the  warmer  water  at  the  bottom.  These  convection  currents  con- 
tinue until  the  water  is  cooled  throughout  to  a  temperature  of  4° ; 
beyond  this  point  the  water  at  the  surface  expands  as  it  cools 
and  remains  at  the  top.  Thus  the  water  at  the  surface  freezes 
while  that  below  remains  at  4°,  even  in  the  most  severe  winters, 
—  a  temperature  at  which  fishes  and  other  inhabitants  of  the 
waters  are  not  destroyed. 

Laboratory  Exercise  J2. 

237.  Expansion  of  Gases.  —  The  effect  of  heat  upon  the  vol- 
ume of  gases  was  first  accurately  investigated  by  the  French  physi- 
cist, Jacques  Charles,  who  discovered  the  law  that  bears  his  name. 

Law  of  Charles  :  The  volume  of  any  gas  increases  under  constant 
pressure  by  -^-^  of  its  volume  at  zero  (Centigrade)  for  each  rise  of 
temperature  of  one  degree. 

The  fraction  ^|^,  or  .003665,  is,  according  to  the  law,  the  co- 
efficient of  cubical  expansion  for  all  gases  at  all  temperatures, 
under  any  constant  pressure.  I^ter  and  more  accurate  experi- 
ments have  shown  that  this  law,  like  that  of  Boyle  (Art.  47),  is 
only  approximately  true ;  though  very  nearly  so  indeed,  unless  the 
gas  is  near  the  temperature  at  which  it  liquefies. 

It  can  be  shown  by  experiment  or  proven  from  the  laws  of 
Boyle  and  Charles  that,  when  a  gas  is  heated  without  being  per- 
mitted to  expand  (volume  constant),  its  pressure  increases  at  the 
same  rate  as  the  volume  does  when  it  is  heated  under  constant 
pressure. 


i8o  Heat 

It  is  instructive  to  contrast  the  identical  behavior  of  all  gases  as 
expressed  by  the  laws  of  Boyle  and  Charles  with  the  very  marked 
individual  differences  exhibited  by  solids  and  liquids  in  their 
relation  to  changes  of  pressure  and  temperature.  The  gaseous 
state  is  evidently  very  much  simpler  than  the  other  two  ;  which  is 
explained  by  the  fact  that  the  molecules  of  a  gas  are  separated 
beyond  the  range  of  cohesion, 

238.  Absolute  Temperature  and  Absolute  Zero.  —  Let  ?'„  be  the 
volume  of  a  body  of  gas  at  o°  C,  and  z^,  its  volume  at  any  other 
temperature  /|  under  the  same  pressure. 

The  increase  in  volume  is  7'i  — rv.  and  this  increase  is  — ^  of 
the  volume  at  0**  (law  of  Charles)  ;  that  is,  —  ^^^ 

From  which  Ti  =  ?'„(  i  H — ~  ).  (i) 

\       273/ 

Similarly,  if  r,  be  the  volume  of  the  gas  at  temperature  /j,  under 
the  same  pressure,  then  — 

.,=..(.+-4^).  (.) 

Dividing  the  members  of  equation  (i)  by  the  corresponding 
members  of  equation  (2),  we  have  — 

gi_        273 

273 

which  reduces  to  ^  =  "^  ^^44 '  (s) 

Vt     2734-/2  ^^^ 

The  relation  expressed  by  equation  (3)  has  led  to  the  adoption 
of  a  temperature  scale  whose  degrees  are  the  same  as  those  of  the 
Centigrade  scale  but  whose  zero  is  at  —  273°  C.  This  scale  of 
temperature  is  called  the  absolute  scale,  and  its  zero  the  absolute 
zero.    The  freezing  point  is  273°  Abs.  and  the  boiling  point  373° 


Temperature  and  Expansion  i8i 

Abs.  Any  temperature  on  the  Centigrade  scale  is  changed  to  the 
absolute  scale  by  adding  273.  If  we  let  T  denote  temperatures 
on  the  absolute  scale,  then  71  =273  +  /!,  and  7^2=  273  +  4,  and 
equation  (3)  becomes  — 

M-  « 

Expressed  in  words  the  meaning  of  this  equation  is :  Under 
constant  pressure  the  volume  of  any  body  of  gas  is  proportional  to 
its  absolute  temperature.  This  is  but  another  (and  the  simplest) 
way  of  stating  the  law  of  Charles.  If  this  law  held  for  all*  tem- 
peratures, it  is  evident  that  at  absolute  zero  the  volume  of  any 
body  of  gas  would  be  zero ;  but,  as  before  stated,  the  law  fails  to 
express  the  behavior  of  gases  when  near  the  point  of  condensation, 
and  all  licjuefy  and  even  solidify  before  reaching  absolute  zero. 
When  near  the  point  of  condensation,  the  decrease  of  volume  for 
a  given  fall  of  temperature  is  less  than  that  indicated  by  the  law. 

By  reasoning  based  on  the  relation  of  heat  to  mechanical 
energy,  it  is  proved  that  the  absolute  zero  is  what  its  name 
indicates ;  namely,  the  temperature  at  which  a  body  would  pos- 
sess no  molecular  kinetic  energy,  or  no  heat,  —  the  molecules 
would  be  at  rest.  No  substance  has  yet  been  cooled  to  absolute 
zero;  but  the  temperature  has  been  closely  approached  by  the 
evaporation  of  hydrogen  after  it  has  been  liquefied  (Art.  268). 
By  this  means  hydrogen  has  been  cooled  to  a  temperature 
estimated  at  —  259°C.  or  14°  Abs.,  at  which  temperature  it  is 
frozen.  Air  freezes  at  a  considerably  higher  temperature  and 
boils  at  — 191°  C.  or  82°  Abs. 

PROBLEMS 

1.  (rt)  The  reading  of  a  thermometer  gives  the  temperaturt  of  the  ther- 
mometer. On  what  grounds  do  we  assume  that  the  reading  of  a  thermometer 
in  a  liquid  gives  the  temperature  of  the  liquid?  (^)  Why  do  we  not  take  the 
reading  immediately  on  inserting  a  thermometer  in  a  liquid  to  determine  its 
temperature? 

2.  The  reading  of  a  barometer  is  76  cm.  on  a  certain  day  when  its  tem- 
perature {i.e.  the  temperature  of  the  mercury  in  the  barometer)  is  0°.    What 


1 82  Heat 

would  have  been  the  reading  of  the  barometer  if  its  temperature  had  been 
20^? 

3.  In  accurate  work  the  reading  of  the  barometer  must  be  "  corrected  for 
temperature "  ;  1^.  the  true  height  is  taken  as  the  heij^ht  at  which  it 
would  stand  if  the  temperature  of  the  mercury  were  o*'.  A  barometer  read- 
ing is  75.6  cm.  at  a  temperature  of  22*^ ;   find  the  true  or  corrected  height. 

4.  The  thinner  a  glass  tumbler  is,  the  less  likely  is  it  to  break  when  hot 
water  is  |)Oured  into  it.     Why? 

5.  Why  cannot  an  air  thermometer  be  used  for  measuring  the  lowest 
attainable  temperatures? 

6.  What  will  be  the  volume,  at  75",  o(  a  body  of  air  which,  under  the  same 
pressure,  has  a  volume  of  250  ccm.  at  20*^  ? 

SlUGESTlON.  —  Use  equation  (3)  or  (4)  above. 

7.  A  body  of  gas  at  10*'  and  a  pressure  of  one  atmosphere  is  inclosed 
in  a  vessel  and  heated  to  300*^.  What  is  the  pressure  at  that  temperature, 
none  of  the  gas  being  allowed  to  escape? 

8.  A  quantity  of  gas  is  found  to  have  a  volume  of  800  ccm.  at  20"  under 
atmospheric  pressure  when  the  barometer  reads  75  cm.  What  would  be  the 
volume  of  the  gas  at  0°  and  a  pressure  of  one  atmosphere? 

Si'G(;estion.  —  Find  the  volume  at  the  given  temperature  and  76  cm. 
pressure  (Boyle's  law),  and  from  this  the  volume  at  0°  under  the  latter 
pressure  (law  of  Charles). 


V.   Calorimetry:  Specific  Heat 

The  Heat  Unit.  —  Heat  being  a  form  of  energy,  it  can  be 
measured  in  terms  of  any  of  the  units  by  which  mechanical  energy 
is  measured  (ft.-lb.,  etc.)  ;  they  are  not  used,  however,  as  there 
are  more  convenient  units  for  the  purpose.  Two  heat  tinits  are  in 
common  use  :  one,  the  calorie^  is  the  amount  of  heat  required  to 
raise  the  temperature  of  one  gram  of  water  one  degree  Centi- 
grade ;  the  other  is  the  amount  of  heat  required  to  raise  the  tem- 
perature of 'one  pound  of  water  one  degree  Fahrenheit.  The 
calorie  is  almost  exclusively  used  in  scientific  work  and  is  the  only 
heat  unit  that  is  used  in  this  book. 

Example.  —  How  much  heat  is  required  to  raise  the  temperature  of  70  g. 
of  water  from  8°  C.  to  63"^  C?  The  rise  of  temperature  is  55°;  hence  the 
heat  required  is  55  calories  per  gram,  and  70  X  55,  or  3850  calories,  for  70  g. 


Calorimetry  :   Specific   Heat  183 

The  amount  of  heat  required  to  raise  the  temperature  of  one 
gram  of  water  one  degree  is  not  exactly  the  same  at  all  tempera- 
tures, but  the  difference  is  too  small  to  be  of  importance  except 
in  the  most  accurate  work.  The  numerical  relation  between  the 
calorie  and  the  units  of  mechanical  energy  is  considered  in  Art.  270. 

240.  Specific  Heat.  —  It  is  found  by  experiment  that  only  one 
ninth  as  much  heat  is  required  to  cause  a  given  rise  of  temperature 
in  any  mass  of  iron  as  is  necessary  to  cause  the  same  rise  of  tem- 
perature in  an  equal  mass  of  water.  This  ratio  J,  or  .11,  is 
called  the  specific  heat  of  iron.  The  term  is  also  applied  to  the 
number  of  calories  required  to  raise  the  temperature  of  one  gram 
of  iron  one  degree y  which  is  evidently  .11  calorie. 

The  specific  heat  of  a  substance  is  the  ratio  of  the  quantity  of 
heat  required  to  raise  the  temperature  of  any  mass  of  the  sub- 
stance one  degree  to  the  amount  required  to  raise  the  temperature 
of  an  equal  mass  of  water  one  degree ;  or,  it  is  the  number  of 
calories  required  to  raise  the  temperature  of  one  gram  of  the  sub- 
stance one  degree  (Centigrade). 

ExAMi'LKS.  —  If  the  specific  heat  of  a  substance  is  .04,  to  raise  the  tem- 
perature of  50  g.  of  it  from  2^  C.  to  6"  C,  would  require  50  x  4  X  .04,  or  8 
calories.  The  same  body  in  cooling  from  50°  C.  to  30°  C.  would  give  out 
50  X  20  X  .04,  or  40  calories. 

The  specific  heat  of  water  is  one,  by  definition  ;  it  is  very  large 
compared  with  that  of  most  other  substances,  especially  the 
metals,  and  is  exceeded  only  by  hydrogen.  In  the  following  table 
the  substances  are  named  in  the  order  of  their  specific  heats. 

Table  of  Specific  Heats 


Hydrogen 3.409 

Water i.ooo 

Alcohol  (0°  to  50°)    .     .     .     .0.615 

Ice 0.504 

Steam 0.480 

Afr 0.237 

Marble 0.216 


Aluminum 0.213 

Glass 0.198 

Iron 0.1 13 

Copper 0.095 

Brass 0.094 

Mercury 0.033 

Lead 0.031 


184  Heat 

241.  Measurement  of  Specific  Heat.  —  The  method  generally 
used  for  determining  the  specific  heat  of  a  substance  is  known  as 
the  method  of  mixtures.  It  is  illustrated  by  the  following  example  : 
A  brass  calorimeter  weighing  100  g.  contains  400  g.  of  water  at 
I8^  Into  this  is  put  a  roll  of  sheet  iron  at  100°,  weighing  190  g. 
After  stirring,  the  temperature  of  the  water  is  22°,  and  this  is 
assumed  to  be  the  temperature  of  the  roll  of  iron  and  the  calo- 
rimeter. The  specific  heat  of  the  calorimeter  is  given  as  .094. 
Find  the  specific  heat  of  the  roll  of  iron. 

Solution.  —  I^t  j  denote  the  specific  heat  of  iron  ;  i.e.  in  this  case,  it  is 
the  number  of  calories  of  heat  given  out  by  each  gram  of  the  roll  of  iron  in 
cooling  one  degree. 

Rise  of  temp,  of  calorimeter  and  water  =  22"  —  18**  =  4° 

Heat  received  by  the  calorimeter  =  100  x  4  X  .094  =        37.6  cal. 
Heat  received  by  the  water  =  400  x  4  =    1600    cal. 

Fall  of  temperature  of  the  iron  =  100°  —  22°  =         78° 

Heat  given  out  by  the  iron  =  190  x  78  X  .f  =  14820  j  cal. 

Assuming  that  the  transfers  of  heat  take  place  only  among  the  calorimeter 
and  its  contents,  it  follows  that  the  heat  given  out  by  the  roll  of  iron  in  cool- 
ing to  the  temperature  of  the  "  mixture  "  is  equal  to  the  heat  gained  by  the 
calorimeter  and  water  in  coming  to  the  same  temperature  ;  that  is, 

14820  J  =  37.6  +  1600, 
from  which  s  —  1637.6  -4-  14820  =  .110. 

242.  The  Heat  Equation. — The  above  example  illustrates  the 
method  of  treating  the  experimental  data  in  all  experiments  in 
calorimetry.  The  following  summary  of  the  method  will  therefore 
be  of  service  now  and  later. 

(i)  Find  numerical  or  algebraic  expressions  for  the  separate 
quantities  of  heat  involved  in  the  equalization  of  temperatures. 

(2)  With  these  quantities  of  heat  form  the  heat  equation^  which 
expresses  the  equality  of  heat  lost  and  heat  gained. 

(3)  The  heat  equation  contains  as  an  unknown  quantity  the 
quantity  sought  (specific  heat,  heat  of  fusion,  or  heat  of  vaporiza- 
tion). To  find  this  quantity,  solve  the  equation  by  the  usual 
algebraic  processes. 


Calorimetry  :  Specific  Heat  185 

243.  The  Control  of  Heat  in  Calorimetric  Experiments.  —  Any 
transfer  of  heat  between  the  contents  of  the  calorimeter  and  the 
surrounding  air  or  other  bodies  during  an  experiment  is  a  source 
of  error,  and  must  be  avoided  in  so  far  as  possible.  The  calo- 
rimeter is  usually  nickel  plated  and  brightly  polished  to  dimin- 
ish radiation  when  it  is  warmer  than  the  surrounding  air,  and  to 
diminish  absorption  when  it  is  cooler.  The  calorimeter  should 
stand  on  a  poor  conductor  (wood)  and  should  be  touched  with 
the  hands  as  little  as  possible,  to  avoid  conduction. 

At  the  beginning  of  an  experiment  the  water  should  be  taken  at 
such  a  temperature  that  it  (and  the  calorimeter)  will  be  colder  than 
the  air  during  a  part  of  the  time  and  warmer  during  a  part,  in 
order  that  the  gain  of  heat  by  conduction  and  radiation  at  the 
lower  temperature  may  be  as  nearly  as  possible  equal  to  the  loss 
by  the  same  means  at  the  higher  temperature. 

Laboratory  Exercise  jy. 

PROBLEMS 

1.  The  specific  heat  of  water  is  much  greater  than  thai  of  rocks  and  soils. 
How  does  this  in  part  account  for  the  fact  that  the  change  of  temperature  of 
the  land  between  day  and  night  and  between  winter  and  summer  is  much 
greater  than  that  of  the  ocean  ? 

2.  Are  equal  quantities  of  heat  required  to  raise  equal  volumes  of  different 
substances  through  equal  changes  of  temperature  ?  (Consult  table  of  densi- 
ties and  table  of  specific  heats.) 

3.  What  effect  has  the  large  specific  heat  of  water  on  the  sensation  caused 
by  putting  the  hand  in  hot  or  cold  water  ?  In  general,  how  does  the  specific 
heat  of  a  substance  affect  the  sensation  of  heat  or  cold  caused  by  it  when 
touched  (see  Art.  217)  ? 

4.  A  roll  of  lead  weighing  800  g.  is  heated  to  100°  and  placed  in  a  brass 
calorimeter  weighing  90  g.  and  containing  406.3  g.  of  water  at  16.2°.  The 
final  temperature  is  21°.     Find  the  specific  heat  of  the  lead. 

5.  A  kilogram  of  mercury  at  200°  and  a  kilogram  of  water  at  0°  are  mixed. 
Find  the  resulting  temperature,  no  allowance  being  made  for  the  vessel. 

6.  A  piece  of  aluminum  weighing  60  g.  is  heated  to  63°,  and  placed  in  a 
copper  calorimeter  weighing  50  g.  and  containing  100  g.  of  alcohol  at  8". 
The  temperature  of  the  alcohol  rises  to  17°.  Find  its  specific  heat,  taking  the 
specific  heat  of  copper  and  aluminum  from  the  ^ble. 


1 86  Heat 

VI.  Fusion  and  Solidification 

244.  Melting  of  Ice  and  Freezing  of  Water.  —  \\lien  heat  is 
applied  to  ice  at  any  temperature  below  o^,  the  temperature  of 
the  ice  rises,  but  melting  does  not  begin  until  the  temperature 
has  risen  to  o*.  With  the  further  application  of  heat  the  ice 
begins  to  melt,  but  its  temperature  remains  at  o°. 

When  a  vessel  of  water  is  surrounded  by  any  substance  whose 
temperature  remains  below  zero,  the  water  loses  heat  and  cools 
U>  ©•.  With  further  loss  of  heat,  the  water  begins  to  freeze ;  but 
its  temperature  remains  at  o°  until  it  is  all  frozen. 

Thus  the  wultimg  point  of  ice  and  the  freeung  point  of  water 
are  exactly  the  same.  Whether  melting  or  freezing  will  take 
place  in  a  >*essel  containing  both  ice  and  water  depends  upon 
whether  heat  is  passing  into  the  vessel  or  from  iL  K  the  water  is 
losing  heat,  it  will  freeze  ;  if  the  ice  is  receiving  heat,  it  will  melt ; 
if  there  is  neither  gain  nor  loss  of  heat,  neither  melting  nor  freezing 
win  occur. 

Lmhoraifrx  Exrrcisf  jj. 

945.  lldting  Points.  —  Every  solid  that  can  be  melted  has  a 
constant  melting  point,  which  is  also  the  temperature  at  which  it 
freezes  or  solidifies.  .Among  fusible  solids,  some,  like  ice,  change 
abruptly  from  the  solid  to  the  liquid  state.  In  such  cases  the  melt- 
ing point  can  be  very  accurately  determined.  Other  solids,  as  seal- 
ing wax  and  glass,  gradually  soften  and  pass  by  continuous  change 
into  the  liquid  state.  In  such  cases  the  melting  point,  although 
constant,  is  nx>re  or  less  indefinite. 

Talflt  of  Melting  Points 

yrfc 

4» 

657 

, 1050 

Gla» 1000  to  1400 

Iron,  wrought    .     .    1500  to  1600 
Flatinom 1775 


-ijo  C 

Moc«y -39 

Ice o 

Batter n 

Beeswax 62 

Caae.ssgar 170 

Solder,  soft 225 


Fusion  and  Solidification  187 

246.  Change  of  Yolnme  during  Fusion  and  Solidification. — 
Most  substances  expand  in  melting  and  contract  in  solidifying; 
in  many  cases  the  change  in  volume  is  considerable.  This  is  well 
illustrated  in  cooling  a  dish  of  melted  beeswax  or  paraffine :  the 
contraction  leaves  a  depression  at  the  center  of  the  cake  {Exp.). 
Metals,  with  few  exceptions,  also  contract  in  solidifying.  Those 
that  do  are  not  adapted  for  casting,  as  they  would  shrink  away 
from  the  surfaces  of  the  mold.  Cast  iron  and  type  metal,  which 
is  an  alloy  of  lead,  tin,  and  antimony,  are  among  the  exceptions. 

Water  expands  in  solidif>ing,  as  is  well  known,  the  increase  in 
volume  amounting  to  about  one  eleventh.  As  a  result  of  ihb 
expansion  ice  floats  —  a  fact  of  great  importance  in  nature.  If 
water  contracted  in  freezing,  ice  forming  at  the  surface  of  lakes 
and  rivers  would  sink.  Freezing  would  consequently  continue  at 
the  surface  throughout  winter  or  until  the  lakes  and  rivers  were 
frozen  solid,  and  all  animal  life  inhabiting  them  would  be  de- 
stroyed. 

The  enormous  force  exerted  by  water  in  freezing  is  shown  in 
the  occasional  bursting  of  water  pipes  in  winter.  The  magnitude 
of  this  force  was  strikingly  shown  by 
some  experiments  of  Major  Wil- 
liams, in  Canada.  "  Having  quite 
filled  a  thirteen-inch  bombshell  with 
water,  he  firmly  closed  the  touchhole 
with  an  iron  plug  weighing  three 
pounds,  and  exposed  it  in  this  state 
to  the  frost.  After  some  time  the 
iron  plug  was  forced  out  with  a  loud 
explosion,  and  thrown  to  a  distance  *^ 

of  415  feet,  and  a  cylinder  of  ice  8  inches  long  issued  from  the 
opening.  In  another  case  the  shell  burst  before  the  plug  was 
driven  out,  and  in  this  case  a  sheet  of  ice  spread  out  all  round 
the  crack."     (Fig.  123.)  — Ganot*s  EUnunts  of  Physics. 

"  Much  of  the  destruction  of  rocks  which  is  taking  place  on  the 
earth's  surface  is  due  to  the  same  quiet  but  intensely  powerful 


i88 


Heat 


action  of  freezing  water.  Rain  sinks  into  the  cracks  and  pores 
which  all  rocks  are  liable  to  contain,  and  when  it  freezes  there,  the 
crack  is  inevitably  widened  or  the  stnicture  of  the  rock  loosened. 
Thus  room  is  made  for  more  water,  which  acts  in  the  same  way 
when  it  freezes ;  and  so  by  degrees  immense  masses  of  rock  and 
earth  are  loosened  from  the  mountainside,  nor  does  the  action  end 
until  the  material  is  reduced  to  the  finest  soil."  —  Madan's  Heat, 
Substances  that  expand  in  solidifying  have  a  crystalline  structure 
in  the  solid  state.  The  crystalline  structure  is  plainly  seen  in  the 
ice  that  first  forms  when  water  begins  to  freeze,  in  the  frost  that 
gathers  on  window  panes,  and  in  snow.  In  crystalline  solids  the 
molecules  are  arranged  in  clusters  of  a  definite  shape,  and  hence 
occupy  more  space  than  when  they  lie  loosely  side  by  side  in  the 
liquid  state  ;  just  as  a  number  of  bricks  would  occupy  more  space 
if  arranged  in  patterns  than  if  packed  in  layers. 

247.  Change  of  Melting  Point  produced  by  Pressure.  —  Experi- 
ments have  shown  that  increase  of  pressure  upon  a  solid  that  ex- 
pands in  melting  tends  to  prevent  melting  by  raising  the  melting 
point.  The  pressure  evidently  opposes  melting  because  it  opposes 
the  expansion  that  accompanies  the  process. 

Similarly,  pressure  upon  a  liquid  that  expands  in  solidifying 
tends  to  prevent  solidification  by  towering  the  melting  point,  for 

in  this  case,  also,  pressure 
opposes  the  expansion  that 
accompanies  the  change  of 
state.  While  lowering  the 
melting  point  opposes  solid- 
ification, it  aids  melting. 
Thus  ice  has  been  melted  at 
—  1 8°  by  a  pressure  esti- 
mated at  several  thousand 
atmospheres.  The  change 
in  the  melting  point  of  ice 
Fig.  X24.  (jue   to   a   pressure   of   one 

atmosphere  would  escape  detection  by  means  of  the  thermom- 


Fusion  and  Solidification  189 

eters  used  in  elementary  physics ;  yet  the  effects  produced  under 
certain  conditions  by  small  changes  of  pressure  are  very  striking. 
For  example,  a  loop  of  fine  wire  to  which  weights  are  attached 
slowly  descends  through  a  block  of  ice  round  which  it  has  been 
passed  (Fig.  124)  ;  yet,  after  it  has  passed  completely  through,  the 
ice  is  in  one  solid  piece  as  at  the  beginning  {Exp.).  The  pressure 
of  the  wire  very  slightly  lowers  the  melting  point  of  the  ice  immedi- 
ately beneath  it ;  and  the  ice  melts,  receiving  the  heat  necessary 
for  the  purpose  from  the  water  filling  the  space  just  above  the 
wire.  This  water  freezes  in  losing  heat,  since  it  is  relieved  from 
pressure.  The  process  is  continuous :  the  water  from  the  ice 
melting  below  the  wire,  passes  round  and  freezes  above  it.  The 
three  stages  of  the  process  are  (i)  melting  under  pressure, 
(2)  change  of  position  of  the  water,  (3)  regelation  (refreezing) 
under  diminished  pressure. 

This  explains  the  hardening  of  snow  into  a  solid  mass  in  making 
snowballs,  and  the  freezing  of  ice  to  flannel  wrapped  around  it. 
The  flow  of  glaciers  is  supposed  to  be  due  to  the  same  action  under 
great  pressure. 

248.  Heat  of  Fusion.  —  We  have  seen  that  ice  melts  only  while 
receiving  heat  at  0°,  and  water  freezes  only  while  losing  heat  at  0°. 
The  quantity  of  ice  melted  or  of  water  frozen  is  proportional  to 
the  gain  of  heat  in  the  one  case,  and  to  the  loss  of  heat  in  the 
other.  This  is  true  of  any  substance  that  has  a  definite  melting 
point.  Since  these  transfers  of  heat  during  change  of  state  take 
place  without  change  of  temperature,  it  is  evident  that  heat  is  lost 
in  the  process  of  fusion  and  is  recovered  during  solidification.  In 
what  form  does  this  energy  exist  in  the  liquid  ? 

A  solid  in  melting  must  receive  energy  in  the  form  of  heat  to 
overcome  (in  part)  the  cohesion  of  its  molecules  (Art.  189). 
After  doing  this  work  the  energy  no  longer  exists  as  heat,  but  as 
potential  energy  in  the  changed  molecular  condition  of  the  sub- 
stance,—  it  is  molecular  potential  energy.  The  process  may  be 
illustrated  by  pulling  apart  two  balls  connected  by  a  rubber  band, 
the  balls  representing  molecules  and  the  tension  of  the  rubber 


190  Heat 

band,  cohesion.  Work  is  done  in  separating  the  balls ;  the  result 
is  potential  energy,  which  is  recovered  when  the  balls  are  per- 
mitted to  come  together  again. 

The  number  of  calories  required  to  melt  one  gram  of  any  sub- 
stance is  called  its  Juat  of  fusion  ;^  this  is  also  the  number  of 
calories  given  out  by  one  gram  of  the  substance  in  solidifying. 
Substances  differ  widely  in  their  heats  of  fusion.  The  heat  of 
fusion  of  water  is  much  larger  than  that  of  most  other  substances. 

Table  of  Heats  of  Fusion 


Caloribs 

Ice    .     > 79.25 

Wax 42 

Zinc 28.13 

Silver 21.07 


Calories 

Tin 14.25 

Sulphur 9.37 

Lead 5.37 

Mercury 2.83 


249.  Determination  of  the  Heat  of  Fusion  of  Ice.  —  The  follow- 
ing example  illustrates  the  process  of  finding  the  heat  of  fusion 
of  ice  by  the  method  of  mixtures :  A  quantity  of  dry,  crushed  ice 
weighing  104  g.  is  placed  in  a  brass  calorimeter  weighing  85  g. 
and  containing  220  g.  of  water  at  44°.  The  temperature  after  the 
ice  is  melted  is  5.3°. 

Solution.  —  Let /denote  the  heat  of  fusion  of  ice. 
Fall  of  temperature  of  calorimeter  and  water  =  44  —  5.3  =  38.7° 

Heat  given  out  by  the  calorimeter  =  85  X  38.7  x  .094  =  309.2  cal. 

Heat  given  out  by  the  water  =  220  X  38.7  =  8614.  cal. 

Heat  received  by  the  ice  in  melting  =  104  /cal. 

Heat  received  by  the  ice  water  in  warming  to  5.3°=  104  X  5-3  =  551.2  cal. 
104/+  551.2  =  309.2  +  8514  ; 
/=  79.5  calories. 

Laboratory  Exercise  j8. 

1  According  to  the  caloric  theory,  heat  must  always  remain  heal,  since  it  was 
regarded  as  a  form  of  matter ;  and  the  heat  that  disappears  in  the  process  of  fusion 
(and  vaporization)  was  called  "latent"  (meaning  hidden),  implying  that  it  still 
exists  as  heat,  although  its  presence  cannot  be  detected.  The  only  form  of  energy 
that  is  properly  called  heat  at  all  was  then  called  "  sensible  "  heat  to  distinguish  it 
from  "  latent  "  heat.  The  terms  have  outlived  the  theory  that  gave  rise  to  them  ; 
thus  the  heat  of  fusion  of  ice  is  often  called  the  "  latent  heat  of  water  "  (Art,  210). 


Fusion  and  Solidification  191 

250.  Heat  of  Solution ;  Freezing  Mixtures.  —  Work  is  done  in 
overcoming  cohesion  in  a  solid  when  it  is  dissolved  as  well  as  when 
it  is  melted ;  and  in  many  instances  there  is  direct  experimental 
evidence  that  heat  disappears  in  the  process,  proving  that  this 
work  is  accomplished  by  heat.*  Thus  when  ammonium  chloride 
or  ammonium  nitrate  is  dissolved  in  water,  there  is  a  fall  of  temper- 
ature of  several  degrees  ;  for  the  heat  required  to  dissolve  the  solid 
is  taken  from  the  nearest  available  source;  namely,  the  water. 
Solution  differs  from  fusion  in  that  it  can  take  place  within  a  wide 
range  of  temperature ;  hence  the  temperature  continues  to  fall 
(unless  heat  is  received  from  the  outside)  until  all  the  solid  is 
dissolved  or  until  the  solution  is  saturated. 

A  mixture  of  one  or  more  solids  and  a  liquid,  or  of  two  solids, 
is  called  2i  freezing  mixture  if  the  solution  or  the  liquefaction  of  the 
solids  causes  a  fall  of  temperature  below  zero.  The  following  are 
examples  of  freezing  mixtures  :  — 

(i)  One  part  by  weight  of  ammonium  chloride  and  one  of 
potassium  nitrate  or  ammonium  nitrate,  powdered  together  and 
dissolved  in  two  parts  of  water.     Fall  of  temperature  about  20°. 

(2)  About  5  parts  of  strong  hydrochloric  acid  and  8  parts  of 
powdered  sodium  sulphate.     Fall  of  temperature  about  30°. 

(3)  One  part  of  table  salt  and  2  parts  of  snow  or  crushed  ice. 
Fall  of  temperature  to  about  —  18°.  The  strong  attraction  of  salt 
for  water  causes  the  ice  to  melt  rapidly.  The  heat  required  to 
melt  the  ice  and  to  dissolve  the  salt  is  taken  first  from  the  ice  and 
salt,  then,  by  conduction,  from  surrounding  bodies.  This  freezing 
mixture  is  well  known  from  its  use  in  making  ice  cream. 

(4)  One  part  each  of  crystallized  calcium  chloride  and  snow 
or  crushed  ice.  Fall  of  temperature  to  about  —  40°.  The  attrac- 
tion of  calcium  chloride  for  water  is  stronger  than  that  of  table 
salt,  and  hence  causes  the  ice  to  melt  more  rapidly  and  at  a  lower 
temperature. 

1  When  chemical  action  accompanies  solution,  it  may  result  in  a  rise  of  temper- 
ature, the  heat  generated  by  the  chemical  action  being  greater  than  the  heat  lost 
in  solution. 


192  Heat 


PROBLEMS 

1.  How  much  heat  is  required  to  convert  750  g.  of  ice  at  —  20°  into  water 
at  50^  ? 

2.  How  many  grams  of  ice  at  0°  can  be  melted  by  500  g.  of  water  at  60°  ? 

3.  A  kilogram  of  ice  and  a  kilogram  of  water,  both  at  o"^,  receive  heat  at 
the  same  rale.  What  will  lie  the  temperature  of  the  water  when  the  ice  has 
all  been  converted  into  water  at  0°  ? 

4.  A  piece  of  aluminum  weighing  250  g.  and  heated  to  100°  is  placed  in  a 
dry  cavity  in  a  block  of  ice,  and  melts  63.1  g.  of  the  ice.  Find  the  specific 
heat  of  the  aluminum,  taking  the  heat  of  fusion  of  ice  as  79.25  calories. 

5.  What  purpose  is  served  by  vessels  of  water  placed  in  a  cellar  where 
vegetables  are  stored  or  in  a  greenhouse  on  a  frosty  night  ? 

6.  (o)  Do  freezing  and  thawing  take  place  more  or  less  rapidly  than  they 
would  if  the  heal  of  fusion  of  ice  were  less  ?  (d)  Of  what  importance  is  this 
in  the  economy  of  nature  ? 


VII.   Vaporization  and  Condensation 

251.  Vaporization.  —  The  change  of  a  substance  from  the  solid 
or  liquid  to  the  gaseous  state  is  called  vaporization.  Vaporization 
may  take  place  at  the  free  surface  of  a  licjuid,  or  within  its  mass 
at  the  place  where  heat  is  applied.  In  the  first  case  the  phenom- 
enon is  generally  called  amporation ;  in  the  second  case,  boil- 
ing,  A  liquid  that  evaporates  readily  is  said  to  be  volatile.  The 
gaseous  form  of  a  substance  that  exists  in  the  liquid  or  the  solid 
state  at  ordinary  temperatures  is  generally  called  a  vapor. 

Evaporation  takes  place  at  the  surface  of  most  liquids  at  all 
temperatures,  but  more  rapidly  as  the  temperature  rises.  It  is 
due  to  molecular  motion.  Some  of  the  molecules  of  a  liquid, 
in  their  irregular  motion,  reach  the  surface  with  a  sufficient  up- 
ward velocity  to  carry  them  into  the  space  above,  out  of  the  range 
of  cohesion,  where  they  exist  as  a  gas  or  vapor.  With  a  rise  of 
temperature  the  velocity  of  the  molecules  is  increased,  and  more 
of  them  are  able  to  escape  from  the  liquid  in  a  given  time.  The 
process  is  illustrated  on  the  largest  scale  in  the  evaporation  of 
water  from  Ihe  surface  of  the  oceans,  lakes,  ponds,  and  streams ; 


Vaporization  and  Condensation         193 

as  a  result  of  which,  the  air  always  contains  a  greater  or  less 
amount  of  water  vapor. 

252.  Disappearance  of  Heat  during  Vaporization.  —  It  is  well 
known  that  evaporation  is  a  cooling  process.  A  room  is  appre- 
ciably cooled  by  the  evaporation  of  water  sprinkled  on  the  floor. 
The  skin  is  cooled  by  the  evaporation  of  water  or  perspiration 
from  it.  This  is  especially  noticeable  in  a  draft,  which  causes 
more  rapid  evaporation  by  carrying  the  vapor  away  as  fast  as  it  is 
formed.  The  rapid  evaporation  of  highly  volatile  liquids,  as  alco- 
hol and  ether,  causes  much  greater  cooling. 

The  cooling  effect  of  evaporation  is  explained  as  follows :  A 
liquid  in  vaporizing  increases  enormously  in  volume.*  In  this  ex- 
pansion work  is  done  against  cohesion  and  also  against  external 
pressure,  and  heat  is  transformed  into  molecular  potential  energy. 
In  evaporation,  as  in  solution,  this  heat  is  taken  from  the  nearest 
available  sources  —  first  the  liquid  itself,  then  adjacent  bodies. 

253.  Vapor  Pressure.  —  A  vapor,  like  any  gas,  exerts 
a  certain  pressure  which  is  proportional  to  its  density 
and  increases  with  its  temperature  ;  but  the  behavior  of 
vapors  differs  from  that  of  other  gases  in  important  re- 
spects, as  shown  by  the  following  experiment. 

A  barometer  tube  is  filled  with  mercury  and  set  up  as 
a  simple  barometer.  Ether  is  introduced  into  the  tube 
at  the  bottom,  drop  by  drop,  by  means  of  a  dropping 
tube  or  a  pipette,  care  being  taken  to  let  no  air  enter 
(Fig.  125).  As  the  first  drop  rises  to  the  top  of  the 
mercury  column,  it  instantly  evaporates,  and  the  pressure 
that  it  exerts  as  a  vapor  causes  a  depression  of  the 
column.  The  pressure  of  the  vapor,  expressed  in  centi- 
meters of  mercury,  is  measured  by  the  amount  of  the  - 
depression.  (Why?)  Each  drop  of  ether  evaporates  as  HhMHI 
it  rises  in  the  tube  and  causes  a  further  depression  of  the  '^^^^' 
column.     This  continues,  however,  only  to  a  certain     fig.  125. 

1  A  cubic  centimeter  of  v/ater  forms  1661  ccm.  of  steam  at  100°  and  a  pressure 
of  one  atmosphere. 


1 94  Heat 

point,  beyond  which  the  liquid  ether  accumulates  above  the  mer- 
cury, and  the  column  remains  stationary.  The  ether  vapor  is  now 
saturatfd ;  i.e.  //  cannot  be  made  denser  at  its  present  temperature. 
Before  this  condition  is  reached  the  vapor  is  unsaturated^  and 
hence  exerts  a  less  pressure. 

The  pressure  exerted  by  the  saturated  ether  vapor  is  found  by 
subtracting  the  height  of  the  mercury  in  the  tube  from  the  read- 
ing of  a  barometer.  When  the  tube  is  inclined,  the  space  occu- 
pied by  the  vapor  becomes  smaller;  but  the  vertical  height  of 
the  mercury  column  remains  the  same  as  before,  indicating  that 
the  vapor  pressure  is  unchanged.  This  is  true  even  when  the 
tube  is  inclined  so  far  that  the  space  occupied  by  the  vapor 
almost  disappears.  In  inclining  the  tube,  the  vapor  is  evidently 
not  compressed  and  made  denser ;  for  in  that  case  it  would  exert 
an  increased  pressure.  The  fact  is  that  a  portion  of  the  vapor  is 
liquefied,  and  the  density  of  the  remainder  is  unchanged.  This 
behavior  agrees  with  the  former  statement  that  the  vapor  is  satu- 
rated, and  cannot  be  made  denser  at  its  present  temperature ; 
and,  since  it  cannot  be  made  denser,  it  cannot  be  made  to  exert  a 
greater  pressure.  The  pressure  of  the  saturated  vapor  is  therefore 
the  maximum  pressure  of  ether  vapor  at  its  present  temperature. 

When  the  tube  is  returned  to  the  vertical  position  and  the  ether 
warmed  by  clasping  the  tube  in  the  hands,  the  mercury  descends 
further,  showing  an  increase  of  vapor  pressure  with  rise  of  tempera- 
ture. This  is  partly  due  to  the  heating  of  the  vapor  already  in  the 
tube,  but  chiefly  to  the  evaporation  of  more  ether.  The  saturated 
vapor  is  denser  at  the  higher  temperature.  If  there  were  no  more 
ether  in  the  tube  to  evaporate,  the  heat  of  the  hand  would  cause 
expansion  of  the  existing  vapor,  and  it  would  become  less  dense 
and  unsaturated. 

Similar  results  are  obtained  throughout  when  alcohol  is  substi- 
tuted for  ether  in  the  experiment ;  but  they  are  all  on  a  greatly 
reduced  scale,  for  the  maximum  pressure  of  alcohol  vapor  is  much 
less  than  that  of  ether  at  the  same  temperatures.  With  water  the 
effects  are  very  slight.     At   20°  the  maximum  vapor  pressure  of 


Vaporization  and  Condensation         195 

ether  is  43.28  cm.  (of  mercury),  that  of  alcohol  4.45  cm.,  and 
that  of  water  1.74  cm. 

Laboratory  Exercise  34,  Parts  I  arid  II. 

254.  Laws  of  Vapor  Pressure.  —  The  above  experiment  illus- 
trates the  first  three  of  the  following  laws  of  vapor  pressure  :  — 

I.  At  a  given  temperature  there  is  a  maximum  density  and 
pressure  for  every  vapor,  in  which  condition  the  vapor  is  said  to 
be  saturated. 

Compression  of  a  saturated  vapor  without  change  of  tempera- 
ture causes  a  portion  of  it  to  condense  (liquefy)  ;  but  the  density 
and  pressure  of  the  remainder  are  not  changed. 

II.  At  the  same  temperature  the  maximum  pressures  of  different 
vapors  are  unequal. 

III.  The  density  and  pressure  of  a  saturated  vapor  increase 
with  the  temperature. 

IV.  The  behavior  of  unsaturated  vapors  is  approximately  like 
that  of  gases,  as  expressed  in  the  laws  of  Boyle  and  Charles. 

255.  Mixture  of  Gases  and  Vapors;  Dalton's  Laws. — The 
following  laws  relating  to  mixtures  of  gases  and  vapors  are  known 
as  Dalton's  laws,  from  their  discoverer. 

I.  The  quantity  of  vapor  which  saturates  a  given  space  is  the 
same,  at  the  same  temperature^  whether  this  space  contains  a  gas 
or  is  a  vacmnn. 

II.  The  pressure  of  the  mixture  of  a  gas  and  a  vapor  is  equal 
to  the  sum  of  the  pressures  which  each  would  exert  if  it  occupied 
the  same  space  alone. 

In  a  vacuum  the  evaporation  of  a  volatile  liquid  is  almost  instan- 
taneous. In  the  presence  of  air  or  any  other  gas,  evaporation 
takes  place  much  more  slowly ;  but,  as  is  implied  in  Dalton's 
first  law,  it  will  not  cease  until  any  inclosed  space  above  the 
liquid  contains  as  much  of  the  vapor  as  it  would  if  the  gas  were 
not  present.  The  kinetic  theory  of  gases  accounts  at  once  for  the 
second  law.  The  law  holds  for  the  mixture  of  any  number  of 
vapors  and  gases,  the  most  familiar  example  of  which  is  the 
atmosphere. 


1 96  Heat 

256.  Water  Vapor  in  the  Atmosphere.  —  The  atmosi)here  is  a 
mixture  of  several  gases,  principally  nitrogen  and  oxygen ;  the 
only  other  constituents  of  importance  are  carbon  dioxide  and 
water  vapor.  All  of  the  constituents  of  the  atmosphere  except 
water  vapor  are  practically  constant  in  amount ;  the  latter  varies 
from  an  inappreciable  fraction  to  about  2  per  cent  of  the  whole,  the 
average  amount  being  not  far  from  i  per  cent. 

The  condition  of  the  water  vapor  in  the  air  with  respect  to 
saturation  is  not  in  the  least  affected  by  the  presence  of  the  other 
gases  (Dalton's  first  law),  and  depends  only  upon  its  own  density 
and  temperature  (which,  of  course,  is  the  temperature  of  the  air)  ; 
yet  common  forms  of  expression  seem  to  imply  that  the  presence 
and  condition  of  the  vapor  are  due  to  some  action  of  the  air. 
Thus  when  the  water  vapor  in  the  air  is  saturated,  we  say  that  the 
air  is  saturated  or  that  the  air  has  all  the  moisture  it  can  hold ; 
although,  strictly  speaking,  it  is  the  space  that  has  all  the  water 
vapor  that  it  can  hold  (at  the  given  temperature) .  There  is  per- 
haps no  objection  to  the  use  of  such  expressions  when  their  true 
meaning  is  understood. 

The  air  is  generally  not  saturated ;  it  is  evidently  not  saturated 
whenever  further  evaporation  can  take  place.  Nonsaturated  air 
may  become  saturated  (i)  by  further  evaporation,  (2)  by  a  fall 
of  temperature,  (3)  by  the  two  processes  combined.  Saturation 
results  from  a  sufficient  fall  of  temperature  because  the  density  of 
a  saturated  vapor  is  less  at  lower  temperatures  (Art.  254,  third 
law).  Consequently  when  the  quantity  of  water  vapor  in  the  air  is 
less  than  that  required  for  saturation  at  the  existing  temperature, 
it  is  equai  to  the  amount  required  for  saturation  at  a  definite  lower 
temperature  (called  the  dew-point). 

257.  The  Dew-point.  —  The  temperature  at  which  the  water 
vapor  present  in  the  air  at  any  time  would  be  saturated  is  called 
the  de7u-point  of  the  air  at  that  time. 

When  any  body  of  air  is  cooled  to  the  dew-point,  condensation 
of  water  vapor  begins,  and  continues  as  long  as  the  temperature 
continues  to  fall.     The  moisture  that  gathers  on  the  outside  of  a 


Vaporization  and  Condensation    '     197 

pitcher  of  ice  water  is  a  familiar  illustration.  The  moisture  comes 
from  the  surrounding  air,  which  is  cooled  by  coming  in  contact 
with  the  cold  pitcher,  and  begins  to  deposit  moisture  as  soon  as 
the  dew-point  is  reached.  A  fall  of  temperature  several  degrees 
below  this  point  generally  occurs,  causing  a  considerable  deposit 
which  runs  down  the  sides.  (What  error  is  implied  in  calling  this 
phenomenon  "sweating"  ?) 

The  dew-point  may  be  determined  experimentally  by  putting 
water  in  a  vessel  on  whose  surface  a  thin  film  of  moisture  can 
easily  be  seen  (as  a  nickle-plated  calorimeter),  and  slowly  cool- 
ing the  water  by  means  of  ice  or  a  freezing  mixture  till  the  first 
trace  of  moisture  appears  on  the  vessel.  The  temperature  of  the 
water  when  this  occurs  is  the  dew-point. 

The  dew-point  varies  between  wide  limits.  In  winter  it  is 
often  many  degrees  below  zero.  It  is,  of  course,  lower  than  the 
temperature  of  the  air  unless  the  air  is  saturated  at  the  time ;  and 
it  approaches  the  temperature  of  the  air  as  the  air  approaches 
saturation. 

Laboratory  Exercise 34^  Part  III. 

258.  Humidity. — The  humidity  (or  relative  humidity)  of  the 
air  at  any  time  is  the  ratio  of  the  amount  of  water  vapor  that  it 
contains  at  the  time  to  the  whole  amount  that  would  be  required 
to  saturate  it  at  the  existing  temperature.  This  ratio  is  usually 
expressed  as  a  per  cent.  Thus  the  humidity  of  the  air  is  75  per 
cent  when  it  contains  three  fourths  as  much  water  vapor  as  would 
be  recfuired  to  saturate  it  at  the  time.  The  humidity  of  saturated 
air  is  100  per  cent  by  definition. 

The  dryness  or  dampness  of  the  air  depends  not  only  upon  the 
amount  of  water  vapor  in  it,  but  also  upon  the  temperature ;  in 
other  words,  it  is  determined  by  the  humidity  of  the  air.  This  is 
illustrated  by  the  well-known  fact  that  very  damp,  cold  air  in  a 
room  becomes  dry  when  the  room  is  warmed,  although  there  is  no 
less  vapor  in  the  room  after  the  heating  than  there  was  before. 
The  air  is  drier  because  its  temperature  is  farther  above  the  dew- 
point;  i.e.  its  humidity  has  been  diminished.     For  example:  At 


198     *  Heat 

10°  C.  (50°  F.)  the  maximum  pressure  of  water  vapor  is  .92  cm. ; 
at  20**  C.  (68°  F.)  it  is  1.73  cm.  Hence  when  saturated  air  at  10° 
is  heated  to  20°,  without  change  in  the  quantity  of  vapor  it  con- 
tains, its  humidity  falls  from  100  per  cent  to  about  53  per  cent. 
At  10°  the  air  would  be  disagreeably  moist ;  at  20°  it  would  feel 
rather  dry. 

259.  Laws  of  Evaporation.  —  The  conditions  affecting  the  rate 
of  evaporation  of  a  liquid  may  be  summarized  as  follows  :  — 

I.  TA^  rate  of  evaporation  increases  with  a  rise  of  temperature, 

II.  The  rate  of  evaporation  increases  with  an  increase  of  the 
free  surface  of  the  liquid. 

III.  The  rate  of  evaporation  of  a  liquid  decreases  as  t/te  space 
around  it  approaches  saturation  by  its  own  vapor ^  and  ceases  when 
that  space  is  saturated, 

IV.  The  rate  of  ei^aporation  in  the  open  air  increases  with  a 
more  rapid  change  of  air  about  the  liquid.  Currents  of  air  (winds) 
carry  the  vapor  away  from  the  space  about  the  liquid,  and  the 
stronger  the  currents  are,  the  farther  will  this  space  be  from  satu- 
ration. 

V.  The  rate  of  evaporation  increases  as  the  density  of  the  air  or 
other  gas  surrounding  the  liquid  is  diminished;  in  a  vacuum  it  is 
almost  instantaneous.  Changes  of  barometric  pressure  are  not 
sufficient  to  materially  affect  the  rate  of  evaporation  in  the  open  air. 

It  follows  from  laws  I,  III,  and  IV  that  the  most  favorable  con- 
ditions for  the  rapid  evaporation  of  water  in  the  open  air  are  pres- 
ent on  a  dry,  hot,  windy  day. 

260.  Condensation  of  Water  Vapor  in  the  Atmosphere.  —  Water 
vapor  is  ahuays  invisible.  The  visible  forms  of  moisture  in  the 
atmosphere  —  as  fog,  mist,  clouds,  and  the  so-called  "  steam  " 
near  the  spout  of  a  kettle  in  which  water  is  boiling  —  consist  of 
minute  particles  of  liquid  water,  and  are  the  result  of  the  conden- 
sation that  accompanies  a  fall  of  temperature  after  the  dew-point 
is  reached.  Dew,  frost,  rain,  sleet,  hail,  and  snow  are  forms  in 
which  the  moisture  of  the  air  is  condensed  and  precipitated.  The 
conditions  under  which  the  different  forms  occur  are  as  follows  :  — 


Vaporization  and   Condensation         199 

Dew  is  condensed  water  vapor  coming  from  the  air  immediately 
surrounding  the  body  on  which  it  appears.  It  has  been  found 
by  numerous  experiments  that  surfaces  upon  which  dew  is  form- 
ing are  always  at  least  3°  or  4°  colder  than  the  air  or  dewless 
surfaces.  (Cooling  below  the  temperature  of  the  air  is  due  to 
the  rapid  loss  of  heat  by  radiation.)  When  the  air  is  nearly 
saturated,  it  is  cooled  to  the  dew-point  by  coming  in  contact  with 
such  surfaces,  and  moisture  is  condensed  upon  them.  Dew  forms 
only  at  night,  and  most  abundantly  during  the  latter  part  of  it; 
when,  by  cooling,  the  air  has  become  nearly  saturated.  It  is 
formed  only  on  calm,  clear  nights  ;  for  on  clear  nights  cooling  is 
most  rapid  (Art.  228),  and  it  is  only  on  calm  nights  that  any  por- 
tion of  the  air  remains  long  enough  in  contact  with  the  cold  sur- 
faces to  be  cooled  to  the  dew-point.  Dew  forms  most  abundantly 
on  the  coldest  objects,  which  are  in  general  the  best  radiators  and 
the  poorest  conductors.  Grass,  leaves,  and  boards  are  good 
examples.  A  board  lying  on  the  ground  will  become  wet  with 
dew  when  a  stone  pavement  remains  dry ;  for  the  stone  is  a  good 
conductor  and  receives  heat  from  the  ground,  which  replaces  that 
lost  by  radiation,  hence  its  upper  surface  is  warmer  than  that  of 
the  board. 

When  the  dew-point  is  at  or  below  zero,  condensation  takes 
place  in  the  form  oi frosty  under  conditions  otherwise  the  same  as 
are  necessary  for  the  formation  of  dew.  The  water  vapor  then 
crystallizes  in  the  solid  state  as  it  condenses,  without  passing 
through  the  intermediate  state  of  a  liquid.  (Is  frost  '•*  frozen 
dew"?) 

At  temperatures  above  zero  the  moisture  of  clouds  is  in  the 
form  of  fog  or  mist.  As  the  individual  particles  grow  by  further 
condensation  and  by  uniting  with  one  another,  they  may  become 
too  large  to  be  sustained  in  the  air,  and  will  then  fall  as  rain. 
A  drop  continues  to  grow  by  uniting  with  smaller  particles  that  it 
meets  with  in  falling  through  the  cloud.  Sleet  is  formed  by  the 
freezing  of  the  raindrops  as  they  fall  through  a  layer  of  air  whose 
temperature  is  below  zero. 


200  Heat 

Snow  is  formed  by  the  condensation  of  vapor  in  the  atmospheie 
at  temperatures  below  zero.  Snow  and  frost  are  formed  under 
the  same  conditions  of  temperature  and  humidity,  and  both  have 
a  beautiful  crystalline  structure. 

"  I/aii  is  formed  in  violent  storms,  such  as  tornadoes  and  thun- 
der storms,  where  there  are  strong,  whirling  currents  of  air. 
Hailstones  are  balls  of  ice,  built  up  by  condensing  vapor  as  they 
are  whirled  up  and  down  in  the  violent  currents,  freezing,  melting 
and  freezing  again  as  they  pass  from  warm  to  cold  currents.  For 
this  reason  they  are  often  made  of  several  layers,  or  shells,  of  ice." 
—  Tarr's  Nov  Physical  Geography, 

PROBLEMS 

1.  For  what  two  reasons  <locs  a  licjuid  evaporate  more  rapidly  in  a  wide 
and  shallow  vessel  than  it  does  in  an  unstoppercd  bottle  ? 

2.  Why  is  a  pan  of  water  often  kept  on  a  heating  stove  ? 

3.  (/i)  Why  does  the  breath  often  form  a  visil)le  cloud  on  a  cold  day  ? 
(^)  Is  it  more  likely  to  do  so  when  the  humidity  Is  high  or  low  ? 

4.  The  moisture  from  the  spout  of  a  kettle  of  boiling  water  is  invisible  for 
a  few  inches  beyond  the  spout,  then  for  some  distance  farther  some  of  it 
forms  a  cloud ;  still  farther  it  is  all  invisible  again.     Account  for  these  facts. 

5.  WTiy  does  frost  form  on  the  inside  of  a  window  pane  but  not  on  the 
outside  ? 

6.  Mow  does  the  formation  of  frost  and  snow  prove  that  water  can  exist  as 
a  vapor  at  lower  temperatures  than  it  can  as  a  liquid  ? 

Laboratory  Exercise  j^. 

261/  Boiling.  —  When  fresh  water  is  heated,  dissolved  air  is 
given  off  in  the  form  of  minute  bubbles,  which  begin  to  form  on 
the  sides  and  bottom  of  the  vessel  as  soon  as  the  water  has  become 
slightly  warm.  ITiese  bubbles  often  rise  to  the  surface  in  large 
numbers,  where  the  air  that  they  contain  escapes.  After  the 
water  has  become  hot,  much  larger  bubbles  begin  to  form  at  the 
bottom  where  the  heat  is  applied.  These  are  bubbles  of  steam  or 
water  vapor.  They  rise  rapidly,  but  disappear  before  reaching 
the  surface,  being  condensed  by  the  cooler  water  near  the  top. 
It  is  the  collapse  of  these  first  bubbles  of  steam  that  causes  the 


Vaporization  and   Condensation         201 

singing  of  a  kettle  of  water  shortly  before  it  begins  to  boil.  As 
the  temperature  of  the  water  approaches  100°,  the  bubbles  rise 
higher,  until  finally  they  burst  at  the  surface,  throwing  the  water 
violently  about.  This  formation  and  escape  of  the  bubbles  of 
steam  is  called  boiling. 

262.  Laws  of  Boiling.  —  The  principal  laws  of  boiling,  as  de- 
termined by  experiment,  are  the  following  :  — 

I.  Boiling  begins  at  a  temperature  which  is  invariable  for  each 
liquid  under  the  same  conditions, 

II.  Whatever  the  intensity  of  the  source  of  heat,  the  temperature 
of  a  boiling  liquid  remains  sensibly  constant. 

The  conditions  here  are  similar  to  those  under  which  fusion 
takes  place  :  energy  in  the  form  of  heat  is  required  to  produce 
the  change  of  state  (Art.  266),  and  a  more  abundant  supply  of* 
heat  merely  causes  vaporization  (boiling)  to  take  plate  more 
rapidly. 

III.  77ie  pressure  of  the  vapor  given  off  by  a  boiling  liquid  is 
equal  to  the  pressure  of  the  air  {or  other  gas)  upon  its  free  surface. 

This  may  be  shown  by  measuring  the  vapor  pressure  at  the 
temperature  of  the  boiling  liquid  by  means  of  a  pressure  gauge 
(I.ab.  Ex.  35);  and  is  directly  evident  from  the  fact  that  the 
vapor  makes  room  for  itself  within  the  liquid  against  the  trans- 
mitted pressure  of  the  air,  which  it  must  therefore  be  able  to 
sustain.  At  any  appreciable  depth,  the  vapor  must  exert  a  slight 
additional  pressure  to  balance  that  due  to  the  weight  of  the  liquid. 
A  liquid  cannot  boil  when  the  pressure  upon  its  surface  is  greater 
than  the  pressure  of  its  saturated  vapor  at  the  existing  tempera- 
ture. 

IV.  An  increase  of  pressure  raises  the  boiling  pointy  a  decrease 
of  pressure  lowers  it. 

This  is  a  necessary  consequence  of  the  preceding  law  together 
with  the  third  law  of  Art.  254.  For  an  increase  of  pressure  upon 
the  free  surface  of  a  liquid  requires  an  equal  increase  of  vapor 
pressure  to  enable  the  vapor  to  form  within  the  liquid,  and  an 
increase  of  vapor  pressure  requires  a  rise  of  temperature. 


202 


Heat 


The  lowering  of  the  boiling  point  under  diminished  pressure  is 

readily  shown  by  either  of  the  fol- 
lowing experiments :  A  glass  of 
water  at  30°  to  40°  is  placed  under 
the  receiver  of  an  air  pump  and  the 
air  exhausted.  When  the  pressure 
is  sufficiently  diminished,  the  water 
boils  violently  {Exp,),  Water  is 
boiled  in  a  round-bottomed  flask 
until  the  air  has  been  expelled  by 
the  steam.  It  is  then  quickly  closed 
with  a  tight  cork,  and  inverted  as 
shown  in  Fig.  126.  When  cold 
water  is  poured  over  the  flask,  the 
waier  within  it  boils  violently.  This 
may  be  repeated  till  the  water  in 
the  flask  is  barely  warm.  The  cold 
water  condenses  some  of  the  vapor,  diminishing  its  pressure  {Exp.). 
As  a  result  of  the  diminished  atmospheric  pressure  at  high  alti- 
tudes, the  boiling  p>oint  of  a  liquid  is  considerably  lower  upon  a 
mountain  than  it  is  near  sea  level.  On  the  summit  of  Mont  Blanc, 
for  example,  water  boils  at  84°. 

263.  Boiling  Points.  —  A  liquid  is  said  to  boil  when  bubbles  of 
vapor  formed  by  vaporization  within  its  mass  are  given  off"  at  its 
surface.  ^Vhen  no  pressure  is  mentioned,  the  boiling  point  of  a 
liquid  is  understood  to  mean  the  temperature  at  which  it  boils 
under  a  pressure  of  one  atmosphere.  The  boiling  point  of  a  liquid 
may  also  be  defined  as  the  temperature  at  which^the  pressure  of 
its  saturated  vapor  is  equal  to  one  atmosphere. 


Tabli  of  Boiling  Points 


Ether 35° 

Chloroform 61.2 

Alcohol 78.4 

Water icx) 


Turpentine 160° 

Glycerine 290 

Mercury 357 

Sulphur 448 


Vaporization  and   Condensation         203 


The  following  table  gives  the  pressure  of  saturated  water  vapor 
(and  hence  also  the  pressure  under  which  water  boils)  at  a 
number  of  temperatures,  the  pressure  being  expressed  in  centi- 
meters of  mercury  in  the  first  column  of  pressures  and  in  atmos- 
pheres in  the  last. 


Temperature 

Pressure 

Temperature 

Pressure 

0° 
100 

.46  cm. 

9.20  cm. 

76.00  cm. 

120° 

140 

160 

1.96  atmospheres 
3.58  atmospheres 
6.12  atmospheres 

Laboratory  Exercise  j6. 

264.  Distillation.  —  A  liquid  can  be  separated  from  impurities, 
or  from  nonvolatile  substances  held  in  solution,  by  boiling  it  in  a 
closed  vessel  and  condensing  the  vapor  as  it  passes  off  through  a 


Fig.  127. 

tube  connected  with  the  vessel.  The  process  is  called  distillationy 
and  the  apparatus  a  still.  The  process  may  be  illustrated  by  dis- 
tilling a  solution  of  copper  sulphate  in  water,  using  apparatus 
similar  to  that  shown  in  Fig.   127.     The  vapor  is  condensed  by 


204  Heat 

inclosing  a  portion  of  the  tube  through  which  it  passes  within  a 
larger  tube  in  which  it  is  surrounded  by  a  continuous  supply  of 
cold  water. 

Two  or  more  liquids  whose  boiling  points  differ  by  several  de- 
grees can  be  separated  from  one  another  by  distillation.  When 
such  a  mixture  is  slowly  boiled,  the  vapor  that  passes  off  contains 
a  much  higher  percentage  of  the  more  volatile  constituent  than  the 
liquid  mixture  does.  Some  of  the  less  volatile  liquid  also  passes 
off,  and  complete  separation  can  be  effecteil  only  by  repeated 
distillation.  This  process,  which  is  generally  known  as  fractional 
distiliation^  is  employed  on  a  large  scale  in  separating  the  constitu- 
ents of  crude  petroleum  and  of  coal  tar,  and  in  the  manufacture 
of  distilled  liquors. 

265.  Cooling  by  Expansion.  —  Work  is  done  upon  a  gas  in  com- 
pressing it,  and  by  a  gas  in  expanding  against  pressure.  In  tlie 
first  case,  mechanical  energy  is  transformed  into  heat  and  the  gas 
is  warmed  (Art.  212)  ;  in  the  second  case,  some  of  the  heat  of  the 
gas  is  transformed  into  mechanical  energy  (kinetic  or  potential) 
and  the  gas  is  cooled,  unless  it  receives  an  equal  supply  of  heat 
during  the  expansion. 

The  cooling  of  a  gas  by  expansion  is  often  beautifully  illustrated 
when  the  air  is  exhausted  from  the  receiver  of  an  air  pump.  As 
part  of  the  air  is  removed,  the  expansion  of  the  remainder  causes 
a  fall  of  temperature  to  the  dew-point,  and  some  of  the  water  vapor 
is  condensed,  forming  a  fog  within  the  receiver.  The  fall  of  tem- 
perature may  be  measured  by  a  thermometer  placed  in  the  re- 
ceiver {Exp.^.  A  further  illustration  is  afforded  by  directing  a 
jet  of  air  from  a  tank  of  compressed  air  against  the  bulb  of  a  ther- 
mometer held  a  few  inches  from  the  opening  (Exp.). 

266.  Heat  of  Vaporization.  —  The  Aea/  of  vaporization  of  a 
liquid  is  the  number  of  calories  required  to  vaporize  one  gram  of 
it  at  its  boiling  point.  The  heat  of  vaporization  of  water  is  536 
calories,  and  is  greater  than  that  of  any  other  substance.  Its  value 
for  alcohol  is  209  calories,  for  ether  90  calories,  for  mercury 
62  calories. 


Vaporization  and   Condensation         205 

Heat  is  lost  during  vaporization  by  transformation  in  part  into 
( I )  mechanical  potential  energy  in  causing  expansion  against  atmos- 
pheric pressure  (external  work),  and  in  part  into  (2)  molecular 
potential  energy  in  overcoming  cohesion  during  expansion  (in- 
ternal work).  When  the  heat  thus  transformed  is  supplied  by 
the  liquid  itself  and  by  adjacent  objects,  as  is  often  the  case  in 
evaporation,  a  fall  of  temperature  results  (Art.  252);  in  boiling, 
the  temperature  remains  constant,  the  amount  of  heat  received 
from  the  fire  or  flame  being  equal  to  the  amount  transformed 
(Art.  262,  second  law). 

The  heat  lost  during  vaporization  is  all  transformed  into  heat 
again  when  condensation  takes  place. 

267.  Determination  of  the  Heat  of  Vaporization  of  Water. — 
The  determination  of  the  heat  of  vaporization  of  water  by  the 
method  of  mixtures  is  illustrated  by  the  following  example  :  A 
brass  calorimeter  weighing  65  g.  contains  200  g.  of  water  at  5°. 
Steam  at  100°  is  passed  into  the  water  till  the  temperature  rises 
to  40°.  It  is  found  on  again  weighing  the  calorimeter  and  con- 
tents that  the  weight  of  the  steam  condensed  in  it  is  12.1  g. 

Solution.  —  Let  v  denote  the  heat  of  vaporization  of  water,  —  in  this  case 
the  number  of  calories  given  out  by  i  g.  of  steam  in  condensing  to  water  at 

lOO"*. 

Rise  of  temp,  of  calorimeter  and  water  =  40—5  =  35° 

Heat  received  by  the  calorimeter  =  65  X  35  X  .094  =214  cal. 

Heat  received  by  the  water  =  200  X  35  =  ycxx)  cal. 

Heat  given  out  by  the  steam  in  condensing  to  water  at  100°  =  12.1  z/cal. 

Heat  given  out  by  the  water  from  the  condensed  steam  in  cool- 
ing to  40°  =  1 2.1  X  60  =  726  cal. 
12.1  V  -H  726  =  70CX)  +  214; 
z/  =  536.2  cal. 

I 
Laboratory  Exercise  jg. 

268.  The  Condensation  of  Gases.  —  All  substances  that  exist 
only  as  gases  at  ordinary  temperatures  exist  also  as  liquids  and 
even  as  solids  at  sufficiently  low  temperatures.  Gases  may  be 
condensed  (i)  by  cooling,  (2)  by  pressure,  or  (3)  by  the  two  pro- 


2o6  Heat 

cesses  combined.  Sulphur  dioxide  (the  gas  formed  by  burning 
sulphur)  is  easily  liquefied  under  atmospheric  pressure  by  a  freez- 
ing mixture  of  ice  and  salt,  its  boiling  point  being  —  10.5°.  It  can 
also  be  liquefied  at  15°  by  a  pressure  of  3  atmospheres.  Carbon 
dioxide  can  be  liquefied  at  15®  by  a  pressure  of  about  52  atmos- 
pheres, or  by  cooling  to  —  80°  under  a  pressure  of  one  atmosphere. 

Oxygen,  nitrogen,  air,  and  hydrogen  can  be  liquefied  only  at 
very  low  temperatures,  however  great  the  pressure.  The  neces- 
sary reduction  of  temperature  in  such  cases  is  effected  by  the 
sudden  expansion  of  a  ix)rtion  of  the  compressed  gas,  as  in  the 
manufacture  of  liquid  air,  or  by  the  evaporation  of  a  gas  that  is 
more  easily  liquefied.  Thus  in  the  liquefaction  of  hydrogen  the 
cooling  is  effected  by  the  evaporation  of  liquid  air  (Art.  238, 
end). 

Following  is  a  table  of  boiling  points  of  certain  gases  under  a 
pressure  of  one  atmosphere  :  — 

Boiling  Points  of  Liquefied  Gases 

C.  Abs. 

Hydrogen -243''  30° 

Nitrogen —  194  79 

Air —  191  82 

Oxygen —  184  89 

Carbon  dioxide —    78.2  194.8 

Ammonia —    38.5  234.5 

Sulphur  dioxide —    10.5  262.5 

269.  Cooling  by  Evaporation :  Applications.  —  The  tempera- 
ture of  our  bodies  is  largely  regulated  by  the  evaporation  of  the 
perspiration.  The  process  is  continuous,  although  we  are  con- 
scious of  it  only  when  the  perspiration  is  formed  more  rapidly 
than  it  can  evaporate,  and  hence  accumulates  on  the  skin.  An 
average  of  about  a  quart  of  water  is  evaporated  from  the  skin 
daily ;  and  the  heat  that  this  requires  is  taken  principally  from 
the  body.  The  importance  of  more  abundant  perspiration  when 
the  body  is  subjected   to   high   temperatures  and  during  active 


Vaporization  and   Condensation         207 

exercise  is  evident.  It  is  not  the  perspiration,  however,  but  its 
evaporation  that  takes  heat  from  the  body  and  thus  prevents  a 
dangerous  rise  of  temperature.  Hence  hot  weather  is  especially 
oppressive  and  dangerous  when  the  humidity  of  the  air  is  high. 
In  the  very  dry  atmosphere  of  deserts  there  is  comparatively 
little  danger  of  sunstroke  even  at  a  temperature  of  100°  F.,  for 
evaporation  is  very  rapid. 

The  use  of  ammonia  (not  ammonia  water)  and  carbon  dioxide  in 
the  manufacture  of  artificial  ice  depends  upon  the  absorption  of 
heat  by  these  substances  in  vaporizing  at  low  temperatures. 
Ammonia  is  used  on  the  largest  scale.  It  is  liquefied  under  pres- 
sure in  long  pipes  exposed  to  the  open  air,  where  the  heat  gener- 
ated by  the  condensation  is  permitted  to  escape.  It  is  thence 
pumped  into  coils  of  pipes  immersed  in  a  large  tank  of  strong 
brine,  where  it  vaporizes  under  a  low  pressure,  cooling  the  brine 
in  the  tank  several  degrees  below  zero  and  freezing  cans  of  fresh 
water  placed  in  the  brine.  The  vaporized  ammonia  is  pumped 
from  the  coils  in  the  tank  into  the  condensing  pipes,  where  it  is 
again  liquefied. 

PROBLEMS 

1.  What  is  the  pressure  in  grams  per  square  centimeter  exerted  by  and 
upon  a  bubble  of  steam  forming  at  a  depth  of  15  cm.,  when  the  barometer 
reads  76  cm.  ? 

2.  (a)  Is  rain  water  distilled  water  ?     (3)  Is  it  perfectly  pure  ? 

3.  Mention  the  important  consequences  of  the  great  specific  heat,  heat  of 
fusion,  and  heat  of  vaporization  of  water,  including  the  connection  that  any 
of  these  properties  may  have  with  the  effect  of  the  ocean  on  climate. 

4.  A  room  4  m.  by  5  m.  and  3  m.  high  is  warmed  by  a  steam  heater. 
Assuming  no  loss,  what  weight  of  steam  must  be  condensed  in  the  heater  to 
warm  the  room  from  10°  C.  to  18°  C.  ?  (Density  of  the  air  1.25  g.  per  cdm.; 
specific  heat  of  air  =  .237.) 

5.  Water  kept  in  porous  earthenware  jars  in  warm  weather  remains  several 
degrees  below  the  temperature  of  the  air.     Explain. 

6.  What  quantity  of  heat  is  required  to  convert  850  g.  of  ice  at  —  20°  into 
steam  at  ICX)°  ? 

7.  How  much  heat  is  given  out  by  500  g.  of  steam  at  100"  in  condensing 
and  cooling  to  water  at  30°  ? 


2o8 


Heat 


Fig.  12 


VIII.  Mutual  Translormations  of  Heat  and  Other  Forms  of  Energy 

270.  The  Mechanical  Equivalent  of  Heat.  —  The  numerical  re- 
lation between  heat  ;>n<l  mechanical  energy  was  first  determined 
by  Dr.  Joule   (Art.  211).    The  principle  of  his  most  successful 

method  is  illustrated  by 
the  simplified  apparatus 
shown  in  Fig.  128.  A 
set  of  paddles  attached 
to  a  vertical  axle  is  made 
to  rotate  in  a  calorimeter 
filled  with  water,  by  the 
fall  of  a  weight  attached 
to  a  cord  wound  round 
the  axle.  Stationary  pro- 
jections, extending  inward 
between  the  paddles  from 
the  sides  of  the  calo- 
rimeter, prevent  the  water  from  revolving  bodily  with  the  paddles. 
The  work  done  by  the  weight  in  falling  is  converted  into  heat 
within  the  calorimeter  through  the  friction  due  to  the  agitation  of 
the  water.  The  mechanical  energy  thus  transformed  is  measured 
in  gram-centimeters  by  the  product  of  th^  weight  and  the  distance 
through  which  it  descends.  The  number  of  calories  generated  is 
computed  from  the  weight  of  the  water  and  the  calorimeter,  the 
specific  heat  of  the  calorimeter,  and  the  rise  of  temperature. 
Hence,  after  making  necessary  allowances  for  sources  of  error, 
the  equivalent  of  a  certain  number  of  calories  is  obtained  in 
gram-centimeters  of  mechanical  energy ;  from  which  the  equiva- 
lent of  one  calorie  is  computed.  The  latter  is  called  the  mechanical 
equwaUnt  of  heat. 

The  value  now  accepted  for  this  equivalent,  after  repeated 
determinations  by  different  experimenters,  is  42,800  g.-cm.;  i.e. 
42,800  g.-cm.  of  mechanical  energy  will  generate  one  calorie  when 
transformed  into  heat,  and  vice  versa. 


Transformations  of  Energy  209 


PROBLEMS 

1.  Compute  the  rise  in  temperature  of  the  water  in  the  calorimeter  of 
Joule's  apparatus  (Fig.  128)  from  the  following  data  :  Mass  of  the  weight 
used  =  30  kg.  Distance  through  which  the  weight  descends  =  25  m. 
Weight  of  water  in  the  calorimeter  =  2  kg.  Weight  of  the  copper  calo- 
rimeter =  I  kg. 

2.  (a)  A  mass  of  iron  weighing  i  kg.  falls  upon  a  stone  from  a  height  of 
100  m.  How  much  heat  is  generated,  assuming  that  the  energy  is  all  trans- 
formed into  heat  ?  (d)  If  half  of  the  heat  is  generated  in  the  mass  of  iron, 
what  is  its  rise  of  temperature  ? 

3.  From  what  height  must  a  mass  of  iron  fall  in  order  that  the  heat  gener- 
ated when  it  strikes  shall  be  sufficient  to  raise  its  temperature  one  degree, 
assuming  that  the  energy  is  all  transformed  into  heat  in  the  iron  itself? 

4.  A  lead  bullet  strikes  a  target  with  a  velocity  of  300  m.  per  sec. 
Assuming  that  90  per  cent  of  its  energy  is  transformed  into  heat  in  itself, 
what  is  its  rise  of  temperature? 

271.  The  Conservation  of  Energy.  —  All  the  examples  of  the 
transference  and  transformation  of  energy  that  have  been  con- 
sidered serve  to  illustrate  the  fact  that  when  energy  is  lost  by  one 
body  or  in  one  form  it  always  reappears  in  another  body  or  in 
another  form.  The  experiments  of  Joule  and  other  physicists  led 
to  the  view  that  the  energy  always  reappears  in  equivalent  amount. 
In  fact,  since  the  middle^  of  the  last  century,  it  has  been  an  ac- 
cepted principle  that  energy  can  neither  be  created  nor  destroyed. 
This  principle,  which  is  of  the  greatest  scientific  and  practical 
importance,  is  known  as  the  conservation  of  energy.  No  person 
acquainted  with  modern  science  regards  a  "perpetual  motion 
.machine  "  as  a  possibility. 

272.  Transformations  of  Solar  Energy.  —  The  energy  of  winds 
is  directly  traceable  to  the  sun,  for  the  winds  are  due  to  the  un- 
equal heating  of  different  portions  of  the  earth's  surface.  Water 
power  has  the  same  ultimate  source ;  for  it  is  by  means  of  energy 
from  the  sun  that  water  is  evaporated  from  the  oceans  and  carried 
through  the  agency  of  winds  to  the  highest  plateaus  and  moun- 
tains. 


2IO  Heat 

The  leaves  of  plants  absorb  carbon  dioxide  from  the  air,  and, 
under  the  action  of  sunlight,  separate  it  into  its  constituents 
(carbon  and  oxygen).  The  carbon  unites  with  water  sent  up  from 
the  roots,  forming  starch  ;  the  oxygen  is  given  off  to  the  air  again. 
From  the  starch  and  various  earthy  materials  absorbed  with  water 
from  the  soil,  the  different  substances  are  formed  which  are  needed 
for  the  growth  of  the  plant.  The  work  done  in  the  leaves  re- 
quires an  expenditure  of  energy.  This  is  obtained  from  the  sun- 
light absorbed  by  the  green  coloring  matter  in  the  leaves  (the 
chlorophyll).  Radiant  energy  is  thus  transformed  and  stored  up 
as  chemical  potential  energy  in  the  substance  of  the  plant  itself. 
The  energy  of  coal  is  also  stored  solar  energv- ;  for  the  coal  beds 
are  the  remains  of  immense  forests  that  grew  long  before  man  ap- 
peared upon  the  earth. 

Animals,  as  already  noted  (Art.  149),  derive  the  energy  for  all 
their  bodily  activities  from  their  food,  and  hence  originally  from 
the  sun,  whether  the  food  be  of  vegetable  or  of  animal  origin. 

The  sun  is,  in  fact,  directly  or  indirectly,  the  cause  of  nearly  all 
terrestrial  phenomena.  The  only  exceptions  of  any  magnitude 
are  the  tides  and  phenomena  due  to  the  slow  shrinking  of 
the  earth  and  to  the  heat  of  its  interior,  such  as  volcanic  action, 
earthquakes,  the  formation  of  mountains,  etc.  Although  the 
interior  of  the  earth  is  intensely  hot,  the  heat  is  conducted  to 
the  surface  so  slowly  that  its  effect  upon  the  temperature  of 
the  atmosphere  is  inappreciable. 

273.  Amount  of  Solar  Radiation.  —  From  a  law  of  radiation  it 
is  known  that  the  amount  of  energy  radiated  from  any  portion  of 
the  sun's  surface  in  a  minute  is  46,000  times  as  great  as  that 
received  by  an  equal  area  of  the  earth's  surface  in  the  same  time. 
Since  the  latter  has  been  approximately  determined  by  experiment, 
the  rate  at  which  the  sun  is  giving  out  energy  is  known  with  the 
same  degree  of  accuracy.  This  amounts  to  over  100,000  horse 
power  per  square  meter  of  the  sun's  surface,  acting  continuously. 
To  maintain  this  rate  of  radiation  by  combustion  "  would  require 
the  hourly  burning  of  a  layer  of  the  best  anthracite   coal  from 


Transformations  of  Energy  2 1 1 

sixteen  to  twenty  feet  thick  over  the  sun's  entire  surface,  —  a  ton 
for  every  square  foot  of  surfice,  —  at  least  nine  times  as  much  as 
the  most  powerful  blast  furnace  in  existence.  At  that  rate  the 
sun,  if  made  of  solid  coal,  would  not  last  6000  years."  ^  The  earth 
receives  only  about  ^^nn^Ji^^T^T^  ^^  ^^^  ^^^^^  radiation  from  the 
sun. 

274.  Source  of  the  Sun's  Energy.  —  The  source  of  the  sun's 
energy  is  a  question  of  the  greatest  scientific  interest.  A  direct 
answer  not  being  obtainable,  various  theories  have  been  suggested, 
all  of  which  recognize  the  principle  of  the  conservation  of  energy. 
The  sun's  heat  cannot  be  maintained  by  combustion,  for  in  that 
case  it  would  have  been  burned  out  long  ago.  "  Nor  can  it  be 
simply  a  heated  body  cooling  down.  Huge  as  it  is,  an  easy  calcu- 
lation shows  that  its  temperature  must  have  fallen  greatly  within 
the  last  2000  years  by  such  a  loss  of  heat,  even  if  it  had  a  specific 
heat  higher  than  that  of  any  known  substance."^ 

The  only  theory  that  has  met  with  general  acceptance  is  Helm- 
holtz's  theory  of  solar  contraction.  This  is  that  "  the  heat  neces- 
sary to  maintain  the  sun's  radiation  is  principally  supplied  by  the 
slow  contraction  of  its  bulk,  aided,  however,  by  the  accompanying 
liquefaction  and  solidification  of  portions  of  its  gaseous  mass. 
When  a  body  falls  through  a  certain  distance  gradually,  against  re- 
sistance, and  then  comes  tp  rest,  the  same  total  amount  of  heat  is 
produced  as  if  it  had  fallen /r^^/^,  and  been  stopped  instantly.  If, 
then,  the  sun  does  contract,  heat  is  necessarily  produced  by  the 
process,  and  that  in  enormous  quantity,  since  the  attracting  force 
at  the  solar  surface  is  more  than  twenty-seven  times  as  great  as 
terrestrial  gravity,  and  the  contracting  mass  is  immense.  .  .  . 
Helmholtz  has  shown  that  under  the  most  unfavorable  conditions 
a  contraction  in  the  sun's  diameter  of  about  two  hundred  and 
fifty  feet  a  year  would  account  for  the  whole  annual  output  of 
heat."*  At  this  rate,  a  period  of  9000  years  would  be  required 
for  a  total  contraction  sufficiently  great  to  be  detected  by  measure- 
ment with  the  best  astronomical  instruments. 
1  Young's  General  Astronomy. 


212 


Heat 


275.  The  Steam  Engine.  —  Many  illustrations  have  been  given 
of  the  transformation  of  mechanical  energy  into  heat.  Such  trans- 
formations result  in  a  loss  of 
available  energy.  The  op- 
posite transformation  of  heat 
into  mechanical  energy  is 
the  useful  one,  and  is  accom- 
plished by  various  forms  of 
heat  engines,  of  which  the 
steam  engine  is  the  most  im- 
portant type.  Figure  129 
represents  a  horizontal  sec- 


^s 


51 


Fig.  129. 


tion  of  the  cylinder,  r,  and  steam  chest,  //,  of  the  stationary 
engine  shown  in  Fig.  130.  Steam  from  a  boiler  enters  the  steam 
chest  through  the  pipe,  /,  whence  it  is  admitted  mto  the  cylinder  at 
the  ends  alternately,  under  the  control  of  a  sliding  valve,  v,  which 


Fig.  130. 

is  moved  back  and  forth  by  the  rod,  r'.  While  the  steam  is 
entering  at  either  end  of  the  cylinder,  the  other  end  is  con- 
nected by  means  of  the  sliding  valve  with  an  exhaust  pipe  at  e, 


Transformations  of  Energy  2 1  3 

through  which  the  steam  on  that  side  escapes  under  reduced 
pressure  into  the  open  air  or  into  a  condensing  chamber,  where 
it  is  condensed  by  cold  water.  In  the  position  shown  in  the 
figure,  the  steam  enters  at  the  left  end  of  the  cylinder,  and 
pushes  the  piston  and  piston-rod,  r,  to  the  right.  Before  the 
end  of  this  stroke,  the  valve  moves  to  the  left,  shutting  off  the 
steam  from  the  left  end  of  the  cylinder,  and  admitting  it  into 
the  right  end  as  soon  as  the  stroke  is  completed.  The  to- 
and-fro  motion  of  the  piston  rod  causes  the  connecting  rod,  R 
(Fig.  130),  to  turn  the  shaft  S.  The  large  fly  wheel  attached  to 
the  shaft  regulates  the  motion,  which  would  otherwise  be  very 
irregular  in  consequence  of  the  varying  magnitude  and  direction 
of  the  force  exerted  by  the  connecting  rod.  The  valve  rod  is 
operated  by  a  device  on  the  shaft,  to  which  it  is  attached. 

A  locomotive  has  two  engines,  one  at  each  side  of  the  boiler  at 
the  front  end.  The  piston  of  each  of  these  engines  is  attached  by 
means  of  the  connecting  rod  to  the  driving  wheels  on  the  same 
side  of  the  boiler. 

276.  Transformation  of  Energy  by  a  Steam  Engine.  —  The 
valves  of  most  engines  are  capable  of  adjustment  so  that  the  sup- 
ply of  steam  can  be  cut  off  at  any  instant  during  the  stroke.  It  is 
generally  cut  off  before  the  middle  of  the  stroke,  and  often  at 
quarter  stroke  or  even  less.  The  stroke  is  completed  by  the  ex- 
pansion of  the  steam  then  in  the  cylinder.  During  this  expan- 
sion the  temperature  of  the  steam  falls  (Art.  265).  Thus,  if  it  is 
admitted  under  a  pressure  of  6  atmospheres  and  discharged  into 
the  air,  its  temperature  falls  from  about  160°  to  100°  (see  Table, 
Art.  263).  The  heat  lost  by  the  steam  in  expanding  is  the  equiva- 
lent of  the  work  that  it  does  against  the  piston. 

Only  a  comparatively  small  part  of  the  energy  of  the  steam  is 
available  for  doing  the  work  of  driving  the  engine.  The  heat  re- 
quired to  convert  the  water  supplied  to  the  boiler  into  steam  at 
the  temperature  at  which  it  leaves  the  cylinder  of  the  engine  is 
all  lost.  The  efficiency  of  an  engine  is  increased  by  utilizing  the 
expansive  force  of  the  steam  to  the  fullest  possible  extent. 


CHAPTER   IX 

SOUND 

I.  Origin  and  Transmission  of  Sound 

277.  Sounding  Bodies.  —  It  can  be  shown  in  a  number  of  ways 
that  any  sounding  body  is  in  a  state  of  vibration.  Although  the 
motion  is  always  too  rapid  to  follow  with  the  eye,  the  fact  of 
motion  is  nevertheless  often  evident  from  the  appearance  of  the 
body.  For  example,  the  string  of  a  violin  or  other  stringed 
instrument  assumes  a  spindle  shape  and  becomes  indistinct  when 
sounding ;  and  the  prongs  of  a  tuning  fork  present  a  similar  ap- 
pearance. The  vibrations  can  also  generally  be  felt  and,  unless 
very  weak,  can  be  shown  by  their  mechanical  effects.  Thus  a 
shower  of  spray  is  thrown  up  when  the  prongs  of  a  vibrating  fork 
are  dipped  in  water  {Exp.) ;  and  a  pith  ball  or  a  bit  of  cork  tied 
to  the  end  of  a  thread  is  driven  away  when  suspended  so  as  to 
touch  the  edge  of  a  sounding  bell  or  glass  jar  {Exp.). 

278.  The  Vibration  of  a  Sounding  Body.  —  As  the  free  end  of  a 
long,  straight,  steel  spring  is  gradually  shortened  by  clamping  it 
nearer  the  end  in  a  vise,  the  vibrations  become  more  and  more 
rapid,  and  finally  produce  an  audible  sound.  As  the  vibrating 
end  is  further  shortened,  —  and  the  rapidity  of  vibration  thereby 
increased,  —  the  pitch  of  the  sound  rises,  i.e.  the  spring  sounds 
a  higher  tone  {Exp.).  Pitch  will  be  studied  later ;  for  the  present 
we  merely  observe  that  it  is  determined  by  the  rate  of  vibration, 
and  that  the  more  rapid  the  vibration  the  higher  the  pitch  be- 
comes {Exp.  with  siren). 

As  the  length  of  the  free  end  of  the  vibrating  spring  is  increased, 
its  vibration  does  not  change  in  character  ;  it  finally  ceases  to  pro- 
duce an  audible  sound  only  because  its  motion  is  too  slow.     The 

214 


Origin  and  Transmission   of  Sound     215 


Fiu.  131. 


character  of  sound  vibrations  can  therefore  be  studied  by  adjust- 
ing the  spring  to  one  or  two  swings  per  second  {Exp.),  It 
will  then  be  seen  that  the  motion  of 
its  end  is  similar  to  that  of  the  bob  of  a 
pendulum.  Starting  at  D'  (Fig.  131), 
the  motion  is  accelerated  to  Z),  where  it 
is  most  rapid,  and  is  thence  retarded  to 
D".  The  motion  is  maintained  by  the 
elastic  force  of  the  spring,  which,  like  the 
tangential  component  of  the  weight  of  a 
pendulum  bob,  is  greater,  the  greater  the 
displacement  from  the  position  of  rest. 

In  the  study  of  sound,  one  vibration 
includes  a  swing  both  ways  (from  D*  to 
Z>"  and  back  to  /^').  The  amplitude  of  vibration  is  the  extent 
of  motion  on  either  side  of  the  position  of  rest.  The  rate  of 
vibration  is  independent  of  the  amplitude ;  the  vibration  is  there- 
fore regular  or  periodic.  This  can  be  shown  by  counting  the 
number  of  vibrations  with  different  amplitudes  when  the  motion 
is  sufficiently  slow;  but  it  is  proved  for  all  sounding  bodies  by 
the  fact  that  the  pitch  does  not  change  as  the  sound  becomes 
fainter  {Exp.  with  tuning  fork).  The  rate  of  vibration  is  meas- 
ured by  the  number  of  vibrations  per  second. 
Laboratory  Exercise  40. 

279.  The  Vibration  of  a  Tuning  Fork.  —  A  tuning  fork  is  so 
frequently  used  in  sound  experiments  that  it  is  important 
to  know  definitely  what  its  motion  is  when  vibrating.  A 
quick  blow  upon  one  prong  in  the  direction  of  the  other 
sets  both  prongs  in  vibration.  Their  motion  is  always 
toward  each  other  and  from  each  other  in  succession 
(Fig.  132).  This  transverse  vibration  of  the  prongs  is 
accompanied  by  a  vibration  of  the  stem  in  the  direction 
of  its  length  {/ongitudina/  wihrsition) ,  which  can  be  dis- 
tinctly felt  by  placing  the  stem  of  a  sounding  fork  against 


Flo.  132. 


the  teeth. 


2i6  Sound 

280.  The  Transmission  of  Sound.  —  Solid  Media.  —  When  the 
stem  of  a  sounding  fork  is  pressed  against  the  top  of  a  table,  the 
sound  becomes  much  louder  {Exp.).  The  stem  in  vibrating 
strikes  a  rapid  succession  of  blows  upon  the  table,  and  the  impulses 
thus  imparted  cause  the  table  to  vibrate  in  unison  with  the  fork 
—  the  table  becomes  a  sounding  body. 

With  the  stem  of  a  sounding  fork  against  one  end  of  a  wooden 
rod  (meter  stick),  the  sound  instantly  becomes  loud  when  the 
other  end  of  the  rod  is  touched  to  a  table  {Exp.).  The  rod 
transmits  the  vibrations  to  the  table,  or,  as  we  commonly  say,  the 
sound  travels  through  the  rod. 

Any  substance  through  which  sound  travels  is  called  a  medium 
for  the  transmission  of  sound,  or  a  sound  medium  (plural  media). 
The  above  experiment  succeeds  equally  well  with  rods  of  different 
material;  in  fact,  any  highly  elastic  (rigid)  solid  is  a  good  sound 
medium.  The  sound  of  a  distant  train  is  plainly  heard  through 
the  rails,  and  the  tread  of  a  galloping  horse  can  be  heard  for  miles 
through  the  earth  by  putting  the  ear  close  to  the  ground.  On 
the  other  hand,  soft  and  yielding  solids  "  deaden  "  sound,  for  they 
transmit  it  poorly.  Thus  little  or  no  sound  will  be  heard  when  a 
rubber  stopper  or  a  roll  of  cotton  wool  is  placed  between  the 
sounding  fork  and  the  table  {Exp.). 

Liquid  Media.  —  Sound  can  be  transmitted  from  a  fork  to  a 
table  through  a  jar  of  water  or  other  liquid  ;  but,  in  order  that  the 
liquid  may  take  up  the  vibrations  with  suf- 
ficient intensity,  the  area  of  the  vibrat- 
ing surface  in  contact  with  it  must  be 
rather  large.  The  stem  of  the  fork  is 
therefore  firmly  inserted  in  a  hole  in  a 
small  block  of  wood,  and  the  block  touched 
to  the  liquid  {Exp.).  Sounds  made  under 
the  surface  of  a  large  body  of  water 
are  transmitted  through  it  over  long  dis- 
tances. 
Fig.  133.  Gaseous  Media.  —  The  air  is  the  usual 


Origin  and  Transmission  of  Sound     2 1 7 

medium  by  which  sound  vibrations  are  transmitted  to  the  ear. 
That  sound  is  not  merely  transmitted  through  the  air  but  by  it 
is  shown  by  the  following  experiment :  A  loud-sounding  body, 
as  an  electric  bell  (Fig.  133)  or  a  metronome,  is  placed  on  a  soft 
cushion  or  suspended  from  wires  under  the  receiver  of  an  air 
pump,  and  the  air  exhausted.  The  sound  grows  continually  fainter 
as  the  exhaustion  proceeds,  and  becomes  inaudible  if  a  good 
vacuum  is  secured.  The  sound  is  restored  when  air  or  any  other 
gas  is  admitted  into  the  receiver  (Exp.) 

Sound,  unlike  radiant  energy,  is  not  transmitted  through  a  vac- 
uum ;  it  is  transmitted  only  by  elastic  substances  —  solid^  liquid, 
and  gaseous. 

281.  Wave  Motion.  —  A  sounding  body  is  the  center  of  a  peri- 
odic disturbance  consisting  of  impulses  exerted  in  rapid  succession 
upon  any  body  in  contact  with  it,  including  the  surrounding  air ; 
and  these  bodies  serve  as  media  for  the  transmission  of  the  dis- 
turbance. Since  this  disturbance  is  invisible,  it  will  be  helpful  in 
the  study  of  its  nature  and  the  manner  of  its  transmission  to  con- 
sider briefly  a  few  cases  of  visible  motion  that  are  in  some  respects 
like  it. 

When  a  stretched  rubber  tube  or  a  spiral  spring,  three  or  four 
meters  long,  is  struck  a  sharp  blow  near  one  end,  a  distortion  is 
produced  which  travels  rapidly  as  a  wave  to  the  other  end.  By 
tying  strips  of  cloth  to  the  tube  at  different  points,  it  can  be  seen 
that,  as  the  wave  passes  any  point,  that  point  moves  quickly  out 
in  a  direction  at  right  angles  to  the  length  of  the  tube  and  returns 
{Exp.).  The  curved  form  that  we  call  the  wave  is,  in  fact, 
passed  from  point  to  point  along  the  tube  by  the  transverse  vibra- 
tion of  successive  portions  of  the  tube.  A  distortion  of  a  different 
character  is  started  by  stretching  a  portion  of  the  tube  near  one 
end  either  considerably  more  or  less  than  the  remainder  and  sud- 
denly releasing  that  portion.  The  strips  of  cloth  will  now  indicate 
a  to-and-fro  or  longitudinal  vibration  as  the  disturbance  passes 
{F.xfi.). 

The  waves  that  pass  over  a  field  of  grain  when  a  strong  wind  is 


21 8  Sound 

blowing  are  due  to  the  forward  bending  and  springing  back  of 
successive  stalks  of  grain,  a  row  of  stalks  at  right  angles  to  the 
direction  of  the  wind  moving  in  unison.  The  vibration  of  each 
head  of  grain  is  mainly  longitudinal  with  respect  to  the  direction 
in  which  the  waves  travel. 

The  motion  of  the  water  in  transmitting  a  water  wave  is  mainly 
a  transverse  vibration ;  as  is  indicated  by  the  rise  and  fall  of  any 
floating  object  as  the  waves  pass  under  it.  (A  chip  in  a  tub  of 
water  serves  well  for  experiment.)  A  wave  is  not  formed  of  the 
same  water  as  it  travels ;  it  is  the  disturbance  that  travels,  not  the 
water.  When  a  pebble  is  dropped  into  a  pool  of  still  water,  a  train 
of  circular  waves  travels  outward  from  the  point  where  the  pebble 
strikes  the  surface.  The  waves  are  circular  because  the  disturb- 
ance is  transmitted  with  equal  velocity  in  all  directions  over  the 
surface ;  they  are  concentric  because  they  all  originate  at  the  same 
point.  Each  wave  consists  of  a  crest  and  a  trough.  The  length 
of  a  wave  is  the  distance  from  crest  to  adjacent  crest  or  from 
trough  to  trough,  measured  radially^  i.e.  toward  or  from  their 
common  center.  The  waves  rapidly  decrease  in  height  as  they 
travel  outward  because  their  energy  is  transferred  (radially  out- 
ward) to  an  ever  increasing  body  of  water. 

282.  Sound  Waves.  —  Suppose  a  sounding  fork  to  be  held  at 
an  end  of  a  tube  of  indefinite  length.  As  the  nearer  prong  moves 
taivard  the  tube,  it  pushes  against  the  air  immediately  in  front  of 
it.  As  this  body  of  air  is  driven  forward  by  the  advancing  prong, 
it  is  slightly  compressed,  and  hence  expands  on  the  farther  side, 


Fig.  134. 

causing  compression  of  the  air  in  advance  of  it  (Fig.  134).  By 
the  repetition  of  this  process  between  successive  portions  of  the 
air,  the  compression  is  rapidly  transmitted  along  the   tube.     As 


Origin  and  Transmission  of  Sound     219 


the  compression  passes  any  section  of  the  tube,  the  air  particles 
in  that  space  move  forward  in  a  body.  Each  particle  starts  for- 
ward when  the  front  of  the  condensation  strikes  it,  and  stops  as 
soon  as  the  condensation  has  passed. 

As  the  prong  of  the  fork  moves  from  the  tube  during  the 
second  half  of  a  vibration,  the  body  of  air  on  the  side  toward 
the  tube  follows  the  fork  up  in  its  retreat,  and  expands  in  doing 


? 


w 


Fig.  135. 


so  (Fig.  135).  This  causes  air  from  a  greater  distance  to  move 
toward  the  fork  in  restoring  the  density  and  pressure  in  that  re- 
gion. Thus  a  rarefaction  is  transmitted  along  the  tube  by  the 
backward  motion  of  the  air  particles.  The  rarefaction  follows  the 
preceding  compression  (or  condensationy  as  it  is  generally  called), 
and  is  itself  followed  by  the  next  condensation  (Fig.  136). 


Fig.  136. 

A  condeng^M'nq  and  flr^j'^Cfi"^  mrpfhrfinn  together  ^Constitute. 
a  sound_ipave.  _A  sound  wave  is  transmitted,  or  propagated,  by 
a  vibration  of  the  air  particles  —  forward  in  the  condensation  and 
backward  in  the  rarefiiction,  as  indicated  by  the  arrows  in  the 
figures.  This  vibration  is  in  a  straight  line  parallel  to  the  direc- 
tion of  propagation  of  the  wave,  i.e.  it  is  io?igifiidinal ;  its  ampli- 
tude is  very  small  —  generally  much  less  than  that  of  the  sounding 
body. 

A  sound  wave  is  represented  graphically  by  a  curved  line  as 
in  the  lower  part  of  Fig.  136.    The  curve  above  the  straight  line 


220  Sound 

represents  the  condensation,  the  curve  below  the  line  the  rare- 
faction. This  representation  must  not  be  regarded  as  a  picture : 
a  sound  wave  does  not  consist  of  a  crest  and  a  trough  as  a  water 
wave  does. 

283.  Sound  Waves  in  the  Open  Air.  —  When  a  sounding  body 
is  surrounded  by  an  open  body  of  air,  the  waves  travel  outward 
from  it  in  all  directions.  Usually  a  condensation  starts  out  on 
one  side  simultaneously  with  a  rarefaction  on  another.  In  the 
simpler  case  where  the  disturbance  is  the  same  at  the  same 
instant  all  round  the  body  (as  when  a  firecracker  is  exploded  in 
the  aiO,  the  waves  travel  outwar'l   n<  .-.,„.-.•  „/;7V  spherical  shells^ 


FIG.  137. 

represented  in  section  by  Fig.  137.  The  waves  are  spherical 
because  they  are  transmitted  radially  with  equal  velocity  in  all 
directions. 

A  wave  front  is  the  surface  bounding  the  front  of  a  conden- 
sation. Under  the  conditions  just  considered  it  is  a  spherical 
surface.  A  wave  length  is  the  distance,  measured  radially,  be- 
tween adjacent  wave  fronts  or  between  any  corresponding  parts  of 
adjacent  waves. 

A  sound  is  (i)  a  set  or  train  of  sound  waves,  or  (2)  the  sensa- 
tion produced  by  such  a  set  of  waves  through  the  organs  of  hear- 
ing.    In  physics  the  word  is  generally  used  in  the  first  sense. 

284.  Energy  of  Sound  Waves;  Intensity  of  Sound.  —  A  part 
of  the  energy  of  a  sounding  body  is  transformed  into  heat  by 
molecular  friction  within  the  body  itself;  but  most  of  it  is  trans- 
ferred to  the  air  or  other  medium  in  which  sound  waves  are  pro- 


Origin  and  Transmission  of  Sound     221 

duced.  The  more  rapidly  the  energy  is  thus  transferred,  the 
greater  will  be  the  intensity  (or  loudness)  of  the  sound  and  the 
more  quickly  will  the  sound  cease. 

The  rate  of  transference  of  energy  from  the  sounding  body  to 
the  medium  depends  upon  (i)  the  amplitude  of  vibration  of  the 
body,  (2)  the  area  of  the  vibrating  surface,  and  (3)  the  density  and 
elasticity  of  the  medium. 

285.  Effect  of  Amplitude.  —  The  gradual  dying  away  of  the 
sound  of  a  bell,  a  piano  wire,  a  tuning  fork,  etc.,  is  due  to  the 
diminishing  amplitude  of  vibration  as  the  body  approaches  a  state 
of  rest.  With  a  decrease  of  amplitude  the  blows  of  the  vibrating 
surface  against  the  surrounding  air  grow  less  vigorous  and  the 
sound  waves  are  correspondingly  weaker. 

286.  Effect  of  the  Area  of  the  Vibrating  Surface.  —  A  narrow 
vibrating  surface  cuts  through  the  air,  producing  little  effect ;  the 
air  slips  round  it.  A  broad  surface  catches  the  air  and  carries  it 
bodily  forward.  This  explains  why  the  sound  of  a  tuning  fork  is 
very  faint  when  it  is  held  in  the  hand  and  loud  when  it  is  touched 
to  a  table.  In  the  latter  case  the  vibrations  are  transmitted  to 
the  air  almost  entirely  by  the  vibrating  table.  The  music  of  a 
violin  or  guitar  comes  practically  entirely  from  the  body  of  the 
instrument,  and  the  music  of  a  piano  from  the  sounding  board 
on  which  the  wires  are  strung. 

287.  Effect  of  the  Density  and  Elasticity  of  the  Medium.  —  In 
the  experiment  with  the  sounding  body  under  the  receiver  of  an 
air  pump,  it  was  observed  that  the  sound  grows  fainter  as  the 
exhaustion  continues,  /.  e.  as  the  density  of  the  remaining  air  is 
diminished.  The  reason  is  obvious :  there  is  less  matter  in 
motion  in  the  wave  of  rarefied  air,  hence  there  is  less  energy  — 
kinetic  energy  being  proportional  to  the  mass  of  the  moving  body. 

A  number  of  experiments  have  already  shown  that  sound  is 
louder  when  transmitted  through  elastic  solids  than  when  trans- 
mitted through  the  air.  A  sounding  fork  held  first  in  the  hand 
then  against  a  table  is  an  excellent  illustration.  The  rigid  wood 
offers  much  greater  resistance  to  the  blows  of  the  fork  than  the 


222 


Sound 


air  does,  and  hence  receives  a  correspondingly  greater  amount 
of  energy  with  each  vibration.*  Observe  also  that  the  fork  conies 
to  rest  much  more  quickly  when  held  against  the  table  (Exp.), 

288.  Effect  of  Distance  on  the  Intensity  of  Sound.  —  It  is  well 
known  that  a  sound  grows  fainter  with  increase  of  distance  from 
the  sounding  body.  Thfe  principle  of  the  conservation  of  energy 
affords  an  explanation  and  a  definite  statement  of  the  law  of 
decrease. 

As  a  sound  wave  travels  in  the  open  air,  its  distance  from  the 
source  is  the  radius  of  its  surface  (the  spherical  wave  front). 
Now  it  is  proved  in  geometry  that  the  surfaces  of  two  spheres 

(or  equal  fractions  of  their 
respective  surfaces)  are  pro- 
portional to  the  squares  of 
their  radii.  Hence,  since 
the  length  of  a  wave  remains 
constant,  its  volume  and  the 
amount  of  matter  in  it  are 
proportional  to  the  square 
of  the  distance  it  has  trav- 
eled in  the  open  air.  This 
f''°-  '3^-  is  illustrated   in  Fig.   138, 

which  represents  a  section  of  a  spherical  wave  at  a  distance  d^ 
from  the  source  and  again  at  twice  that  distance,  or  //g.  If  Z'l  de- 
notes the  volume  of  the  section  at  the  first  distance,  v>i  its  volume 
at  the  second  distance,  then  z^g :  z'l :  •  4  •  i,  or  z/j :  Z'l : :  //g* :  di. 

Since  the  energy  of  a  sound  wave  is  transmitted  through  the 
medium  with  the  wave,  the  intensity  of  the  sound  (or  the 
amount  of  energy  per  unit  volume)  is  inversely  proportional  to 
the  volume  of  the  wave.  Hence,  from  the  above  proportion,  the 
intensity  is  inversely  proportional  to  the  square  of  the  distance  from 
the  source y  when  the  sound  is  traveling  in  the  open  air. 

1  That  more  work  can  be  done  upon  the  body  that  offers  the  greater  resistance 
is  plainly  evident  to  one  who  strikes  out  at  something  with  his  fist  and  only  suc- 
ceeds in  "  hitting  the  air." 


Origin  and  Transmission   of  Sound     223 

289.  Dissipation  of  the  Energy  of  Sound.  —  We  have  assumed 
that  the  total  energy  of  a  sound  wave  remains  constant  as  it  travels. 
This  is  not  strictly  true,  for  the  energy  is  more  or  less  slowly  trans- 
formed into  heat  by  friction  in  any  medium.  In  the  end  it  is  all 
dissipated  in  this  manner.  Hence  the  intensity  of  sound  decreases 
somewhat  more  rapidly  than  the  law  stated  above  indicates. 

290.  Confined  Sound  Waves.  —  Sound  travels  long  distances  in 
elastic  media  with  very  little  loss  of  intensity  if  the  waves  are  pre- 
vented from  increasing  in  size.  This  is  the  principle  of  the  speak- 
ing tube,  which  is  a  long  metal  tube  of  small  diameter,  used  for 
communication  between  different  rooms  of  a  building  or  between 
some  room  and  the  street  door.  The  tube  does  not  readily  take 
up  the  vibrations  of  the  air,  hence  they  are  almost  completely  con- 
fined within  it.  The  intensity  of  the  sound  is  slowly  diminished 
by  friction. 

The  transmission  of  sound  over  long  distances  through  the  rails 
of  a  track  and  through  stretched  wires  and  strings  is  due  to  the 
same  cause :  the  vibrations  are  largely  confined  to  the  solid 
medium.     This  fact  is  utilized  in  the  acoustic  or  siring  telephone. 

PROBLEMS 

1.  Are  sound  waves  transmitted  by  the  elasticity  of  form  or  of  volume  of 
the  medium  ? 

2.  (fl)  By  what  force  are  the  waves  transmitted  along  a  stretched  rubber 
tube  or  spring  ?  (J))  By  what  force  are  waves  transmitted  over  the  surface 
of  water  ? 

3.  How  does  the  energy  of  sound  differ  from  heat  ? 

4.  Is  the  wave  front  of  a  water  wave  a  line  or  a  surface  ? 

5.  {a)  What  angle  does  the  direction  of  propagation  of  a  water  wave 
make  with  the  wave  front  ?  (Jj)  The  direction  of  propagation  of  a  soupd 
wave  ? 

6.  {a)  Would  a  sounding  body  continue  to  vibrate  longer  in  water  or  in 
the  air  ?     (^)   In  the  air  or  in  a  vacuum  ?     Why  ? 

7.  How  does  the  intensity  of  sound  at  a  distance  of  5  m.  from  the  source 
compare  with  its  intensity  at  10  m.  ?  at  15m.?  at  20  m.  ? 

8.  At  what  distance  is  the  intensity  of  sound  one  fourth  as  great  as  at 
100  ro.  ?.  one  half  as  great  ? 


224  Sound 

291.  The  Velocity  of  Sound  in  Air.  —  It  is  a  familiar  fact  that  a 
distant  phenomenon  that  is  accompanied  by  a  sound  is  seen  before 
it  is  heard.  At  a  distance  of  a  few  hundred  feet,  the  blow  of  an 
ax  is  heard  after  the  ax  is  raised  for  the  next  stroke ;  a  flash  of 
Hghtning  is  often  seen  many  seconds  before  the  thunder  is  heard, 
although  they  are  produced  simultaneously ;  the  whistle  of  a  dis- 
tant locomotive  may  not  reach  the  ear  until  the  cloud  of  "  steam  " 
has  disappeared.  Now  the  time  required  for  light  to  travel  terres- 
trial distances  is  wholly  inappreciable  (the  velocity  of  light  being 
1 86,000  mi.  per  sec.) ;  hence  the  time  that  elapses  between  the 
sensations  of  sight  and  hearing  in  such  a  case  is  the  time  occupied 
by  the  sound  in  traveling  from  the  sounding  body  to  the  observer, 
and,  from  the  measured  time  and  distance,  the  velocity  of  sound 
can  be  computed. 

Observations  have  been  repeatedly  taken  for  this  purpose  by 
firing  a  cannon  at  each  of  two  stations  several  miles  apart,  and 
noting  the  time  between  the  flash  and  the  report  as  observed  at 
the  other  station.  By  taking  observations  at  each  of  the  stations 
alternately,  the  effect  of  the  wind  is  eliminated.  The  average  of 
the  best  determinations  is  332  m.  or  1090  ft  per  sec.  at  0°,  At 
20°  the  velocity  is  344  m.  or  11 29  ft.  per  sec. 

That  the  velocity  of  sound  is  independent  of  its  pitch  and  intensity 
is  proved  by  the  fact  that  all  the  sounds  produced  simultaneously 
by  an  orchestra  are  heard  simultaneously  at  all  distances. 

Laboratory  Exercise  41. 

292.  Effect  of  the  Elasticity  and  Density  of  the  Medium  on  the 
Velocity  of  Sound.  —  Experiment  shows  that  a  wave  travels  more 
rapidly  along  a  stretched  rubber  tube  when  the  tension  is 
increased,  — a  result  to  be  expected,  since  the  propagation  of  the 
wave  is  due  to  the  elastic  force  (tension)  of  the  cord  {Exp.).  The 
elasticity  and  velocity  are  not,  however,  proportional. 

A  wave  travels  more  slowly  along  a  stretched  rubber  tube  that  is 
filled  with  shot  or  sand  than  it  does  along  an  empty  tube  of  the 
same  size  and  under  the  same  tension  —  a  given  force  moves  a 
greater  mass  more  slowly  {Exp.). 


Origin  and  Transmission  of  Sound     225 

It  can  be  shown  by  mathematical  reasoning  based  on  the  second 
law  of  motion  that  the  velocity  of  a  sound  wave  is  proportional 
to  the  square  root  of  the  elasticity  of  the  medium  and  inversely 
proportional  to  the  square  root  of  its  density.  The  velocity  of 
sound  in  water  has  been  found  by  experiment  to  be  1435  "^*  P^^ 
sec.  at  8.1°, —  a  velocity  more  than  four  times  as  great  as  in  air. 
Thus,  in  comparison  with  air,  the  retarding  effect  of  the  greater 
density  of  water  is  more  than  offset  by  the  accelerating  effect  of 
its  still  greater  relative  elasticity.  The  same  is  true  in  a  still  greater 
degree  of  soHds,  the  velocity  in  glass  and  steel  being  about  fifteen 
times  as  great  as  in  air. 

The  velocity  of  sound  increases  with  the  temperature  because, 
with  a  rise  of  temperature,  the  air  expands  and  its  density  dimin- 
ishes, while  its  elasticity  remains  unchanged.  An  increase  of 
pressure  increases  the  elasticity  and  density  proportionally,  hence 
a  change  of  pressure  does  not  affect  the  velocity. 

293.  Reflection  of  Sound :  Echoes.  —  When  sound  waves  strike 
a  large  surface,  as  a  cliff  or  the  side  of  a  building,  they  are 
reflected.  The  reflected  sound  is  called  an  echo  when  it  reaches 
the  ear  long  enough  after  the  original  sound  to  be  distinguished 
from  it.  This  requires  an  interval  not  less  than  \  sec,  during 
which  time  sound  travels  about  68  m. ;  hence  a  distinct  echo  will 
not  be  heard  unless  the  reflecting  surface  is  at  a  distance  not  less 
than  about  34  m.  from  the  source  of  sound.  At  less  distances  the 
direct  and  the  reflected  sounds  blend  together.  They  are  sensibly 
coincident  when  the  reflecting  surface  is  not  more  than  a  few 
meters  from  the  source  of  the  sound,  and  the  result  is  an  increased 
loudness.  It  is  for  this  reason  that  reflecting  surfaces  are  often 
erected  behind  band  stands.  When  the  distance  is  nearly  suffi- 
cient for  an  echo,  the  direct  and  the  reflected  sounds  are  mixed 
confiisedly,  causing  indistinctness.  This  is  often  noticeable  in 
large  halls. 

The  change  in  the  shape  and  the  direction  of  propagation  of 
sound  waves  caused  by  reflection  from  a  plane  surface  is  shown 
in  Fig.  139,  which  represents  a  section  of  a  train  of  waves  origi- 


226 


Sound 


nating  at  (9  and  reflected  by  a  plane  surface,  AB,  The  circular 
arcs  represent  the  wave  fronts,  and  the  lines  OA,  OB,  and 
/)N^    y^  /  .  ^^  directions  of  propagation 

before  reflection  and  AD,  BR^ 
and  CF  directions  after  reflec- 
tion. The  waves  after  reflection 
have  the  same  shape  and  direc- 
tion of  propagation  as  they 
would  have  if  they  originated 
at  0\  a  point  on  the  perpen- 
dicular from  the  source  of  sound 
to  the  reflecting  surface  and  at 
an  equal  distance  behind  it. 
'*'■  *^  After  reflection  from  a  con- 

cave surface,  sound  waves  increase  less  rapidly  in  size,  and  are 
consequently  propagated  with  comparatively  little  loss  of  intensity, 
almost  as  if  they  were  confined  in  an  inclosed  space.  Hence  the 
large  reflecting  walls  at  the  rear  of  band  stands  are  concave. 
When  a  sounding  body  is  at  the  proper  distance  from  a  concave 
surface,  the  reflected  waves  decrease  in  size  and  increase  in  inten- 
sity as  they  travel  toward  a  point.  Light  is  similarly  reflected 
from  concave  mirrors,  such  as  are  used  behind  wall  lamps  and  the 
head  lights  of  locomotives  and  street  cars.  This  efl'ect  of  concave 
surfaces  will  be  more  fully  discussed  under  the  subject  of  light. 
Laboratory  Exercise  42, 


PROBLEMS 

1.  {a)  Give  two  reasons  why  sound  travels  farther  through  the  rails  of  a 
track  than  it  does  through  the  air.  {b)  Why  does  it  travel  faster  through  the 
rails? 

2.  How  would  music  be  affected  if  sounds  of  different  pitch  or  intensity 
traveled  with  different  velocities  ? 

3.  A  rifle  is  fired  on  one  side  of  a  canyon  and  3.2  sec.  later  the  echo  is 
heard  from  the  opposite  side.  The  temperature  is  20°.  What  is  the  width 
of  the  canyoQ  ? 


Properties  of  Musical  Sounds  227 

4.  A  flash  of  lightning  is  seen  12.5  sec.  before  the  thunder  is  heard.  At 
what  distance  did  the  lightning  occur,  the  temperature  being  20°  ? 

5.  The  mean  distance  of  the  sun  from  the  earth  is  93,(XX),C)00  miles.  How 
long  after  an  explosion  occurs  upon  the  sun  would  we  hear  it  if  air  at  0° 
were  provided  as  a  medium  for  the  transmission  ?  (Light  reaches  us  from  the 
sun  in  499  sec.) 


II.  Properties  of  Musical  Sounds 

294.  Properties  of  Musical  Sounds.  —  Musical  sounds  have 
three  characteristics  or  properties  ;  namely,  (i)  intensity  or  loud- 
ness, (2)  pitch,  and  (3)  quality  or  timbre. 

Loudness.  —  Intensity  has  already  been  considered.  It  is  de- 
termined by  the  energy  of  the  sound  waves.  Loudness  refers  to 
the  sensation  produced  upon  the  ear.  It  increases  with  the 
intensity ;  but  we  cannot  say  that  they  are  proportional,  as  loud- 
ness cannot  be  measured.  Loudness  depends  in  part  upon  the 
pitch,  a  shrill  sound  being  more  distinctly  heard  than  one  of 
equal  intensity  but  of  low  pitch. 

Pitch.  —  Sounds  of  definite  pitch  are  produced  only  by  bodies 
whose  vibrations  are  regular  and  periodic.  The  waves  of  such  a 
sound  are  of  equal  length  and  are  sent  out  from  the  sounding 
body  at  equal  intervals  of  time.  An  increase  in  the  number  of 
vibrations  produces  what  is  called  a  higher  pitch  {Exp.). 

Quality  or  timbre  is  that  property  by  which  we  distinguish  be- 
tween sounds  produced  by  different  bodies,  even  when  they  have 
the  same  pitch  and  intensity.  We  know  at  once  from  the  quality  of 
the  sounds  whether  a  piece  of  music  is  being  played  on  a  violin, 
a  flute,  or  a  cornet.  We  recognize  familiar  voices  principally  by 
their  quality,  although  pitch,  loudness,  and  peculiarities  of  pronun- 
ciation are  also  of  assistance.  The  cause  of  quality  will  be  con- 
sidered later. 

295.  Differences  between  Musical  Sounds  and  Noises. — The 
pitch,  intensity,  and  quality  of  musical  sounds  remain  constant  for 
appreciable  intervals  of  time  and  do  not  change  irregularly.  A 
musical  sound  is  often  called  a  tone  or  note.     All  other  sounds  are 


228  Sound 

called  noises.  A  noise  usually  consists  of  a  number  of  sounds  pro- 
duced by  the  vibration  of  the  sounding  body  in  parts  or  segments. 
These  vibrations  not  only  differ  among  themselves,  but  are  also 
irregular  in  rate  and  amplitude ;  they  are  discordant,  unsteady, 
and  nonperiodic.  A  noise  has  therefore  no  definite  pitch,  and 
its  quality  is  nonmusical. 

296.  Measurement  of  Pitch. — The  pitch  of  a  note  may  be 
expressed  either  relatively  or  absolutely.  It  is  expressed  rehi- 
tively  by  stating  its  relation  to  some  other  note,  generally  the 
keynote  of  the  musical  composition  in  which  it  occurs  (Art.  302). 
The  relative  pitch  of  a  note  is  easily  recognized  by  a  trained  ear. 

ThQ  abso/ufe  pitch  of  a  note  is  measured  by  the  number  of 
vibrations  per  second  of  the  sounding  body.  This  is  called  the 
vibration  number  ox  frequency  of  the  note.  Thus  the  absolute 
pitch  of  the  C  fork  which  corresponds  to  "  middle  C  "  of  the 
piano  or  organ  is  256 ;  />.  the  prongs  of  this  fork  make  256 
vibrations  per  second.  In  physics  the  word  pitch  generally 
signifies  the  absolute  pitch  or  vibration  number  of  the  note. 

'llie  vibration  number  of  a  sound  can  be  determined  experi- 
mentally either  by  causing  the  sounding  body  to  make  a  perma- 
nent record  of  its  vibrations  (called  the  graphic  method),  or  by 
comparing  it  with  a  sound  of  known  pitch.     The  graphic  method 

is  illustrated  in  part  in 
Fig.  140.  A  projecting 
point  attached  to  one 
prong  of  the  fork  traces 
f.,^,  a  wavy  line  on  a  piece 

of  smoked  glass  as  the 
vibrating  tbrk  is  drawn  over  the  glass,  or  the  glass  pulled  from 
under  the  fork.  Each  wave  of  the  line  is  a  record  of  one  vibra- 
tion of  the  fork.  If  by  some  additional  device,  not  shown  in  the 
figure,  the  time  occupied  by  the  fork  in  tracing  the  line  is  deter- 
mined, the  number  of  vibrations  per  second  can  be  computed. 
Comparison  with  sounds  of  known  pitch,  such  as  the  notes  of 
tuning  forks,  will  serve  all  the  purposes  of  elementary  physics. 


Properties  of  Musical  Sounds  229 


297.  The  Relation  between  Pitch,  Wave  Length,  and  Velocity.  — 
Let  n  denote  the  vibration  number  of  a  sounding. body,  /  the 
length  of  the  waves  that  it  produces  in  a  given  medium,  and  v 
the  velocity  of  sound  in  that  medium.  Since  a  wave  starts  from 
the  body  with  each  vibration,  a  train  of  n  waves  is  sent  out  in  one 
second,  the  last  of  which  will  be  on  the  point  of  leaving  the  body 
at  the  end  of  the  second.  During  the  second  the  first  wave  of 
the  train  travels  the  distance  v ;  hence,  the  n  waves  extend  over 
that  distance.     From  which  v  =  /n. 

Assuming  the  velocity  of  sound  in  the  medium  to  be  known,  we 
can  from  this  relation  compute  the  wave  length  of  a  sound  of 
given  pitch  or  the  pitch  of  a  sound  of  known  wave  length. 

Laboratory  Exercise  43. 

298.  Interference  of  Sound.  —  When  a  sounding  fork  is  held 
close  to  the  ear  and  is  slowly  rotated  about  the  stem  as  a  vertical 
axis,  it  will  be  observed  that  during  one  complete  rotation  there 
are  four  positions  of  the  fork  in  which  it  is  not  heard.  The  sound 
is  faintly  heard  when  the  fork  is  turned  very  slightly  in  either 
direction  from  a  position  of  silence,  and  swells  to  a  maximum 
midway  between  these  positions.  With  the  fork  held  in  a  posi- 
tion of  silence,  sound  is  restored  by  covering  either  prong  with  a 
small  cylinder  of  paper  or  other  material,  care  being  taken  not  to 
touch  the  prongs,  as  this  would  stop  the  vibration  {Exp.). 

These  curious  effects  are  explained  by  Figs.  141  and  142,  which 
represent  the  sound  waves  about  a  vibrating  fork,  as  seen  with  the 
ends  of  the  fork 
pointing  toward 
the  observer.  As 
the  prongs  move 
apart,  a  conden- 
sation is  set  up 
on  the  outside  of 
each,  and  a  rare- 
faction  between 


them ;    as    they 


Fig.  141. 


Fig.  142. 


230  Sound 

move  toward  each  other,  opposite  conditions  are  produced.  If  tlie 
space  about  the  fork  were  partitioned  off  into  four  compartments, 
as  indicated  in  the  figures,  there  would  be  condensations  and  rare- 
factions on  opposite  sides  of  the  partitions  at  equal  distances  from 
the  fork,  as  shown  in  Fig.  141  ;  but,  without  such  partitions  to 
keep  the  condensations  and  rarefactions  apart,  their  opposing  ten- 
dencies destroy  each  other  where  they  meet,  causing  silence  at 
these  places,  as  shown  in  Fig.  142. 

The  principle  of  the  composition  of  motions  as  studied  in 
mechanics  applies  to  the  resultant  vibration  of  any  portion  of  a 
medium  when  acted  upon  simultaneously  by  two  or  more  trains 
of  waves.  The  waves  from  a  tuning  fork  are  an  example  of  the 
simplest  case ;  namely,  that  of  two  trains  of  waves  of  exactly 
equal  wave  length  and  amplitude,  traveling  in  the  same  direction, 
the  waves  of  one  train  being  half  a  wave  length  in  advance  of 
those  of  the  other  train.  In  the  region  where  the  two  trains  of 
waves  meet,  the  condensations  of  each  unite  with  the  rarefactions 
of  the  other.  The  condensations  would  be  transmitted  by  a  for- 
ward motion  of  the  air,  and  the  rarefactions  by  an  equal  backward 
motion  at  the  same  time;  hence  the  air  in  this  region  remains  at 
rest  and  there  is  neither  condensation  nor  rarefaction.  Silence  is 
thus  the  result  of  the  inUrference  of  the  waves  with  each  other. 

299.  Beats.  —  The  sounds  of  two  forks  of  exactly  the  same 
pitch  unite  perfectly  into  one  sound.  If  the  forks  are  sounded 
together  after  slightly  lowering  the  pitch  of  one  of  them  (by  stick- 
ing a  piece  of  soft  wax  near  the  end  of  one  or  both  prongs),  the 
sound  periodically  swells  and  dies  away  in  strongly  marked  pulsa- 
tions, called  beats  {Exp.), 

Let  us  suppose  that  two  middle  C  forks  are  used  (vibration 
number  =  256),  and  that,  by  loading  with  wax,  one  is  reduced  to 
255.  At  intervals  of  one  second  the  forks  "keep  step"  in  their 
vibrations ;  and  the  waves  that  they  then  set  up  approximately 
coincide  —  condensations  with  condensations  and  rarefactions  with 
rarefactions,  as  at  ^  and  C  (Fig.  143).  These  waves  unite  in 
resultant  waves  of  increased  intensity,  as  represented  at  X  and  Z. 


Properties  of  Musical  Sounds  231 

Half  a  second  after  each  coincidence,  the  forks  vibrate  oppositely, 
the  condensations  produced  by  each  approximately  coinciding 


with  the  rarefactions  produced  by  the  other,  as  at  B;  and  the 
resultant  waves  are  of  diminished  intensity.  A  complete  set  of 
intensified  and  weakened  resultant  waves  is  thus  sent  out  from  the 
forks  in  one  second.     This  constitutes  one  beat. 

Beats  may  therefore  be  defined  as  regularly  recurring  pulsations 
of  sound  caused  by  the  successive  reenforcement  and  interference 
of  two  sets  of  sound  waves  differing  slightly  in  wave  length  or  pitch. 
The  number  of  beats  per  second  is  equal  to  the  difference  between 
the  vibration  numbers  of  the  two  sounds.  Hence  the  beats  become 
more  frequent  when  the  pitch  of  the  loaded  fork  is  further  lowered 
by  using  more  wax  (Exp.), 

PROBLEMS 

1.  Show  from  the  formula  v  =  /n  that  the  wave  length  of  a  sound  of  given 
pitch  is  proportional  to  the  velocity  of  sound  in  the  medium. 

2.  Find  the  wave  length  of  middle  C  in  the  air  at  20*^;   in  water. 

3.  Why  is  the  sound  of  a  fork  restored  in  the  position  of  silence  when  one 
prong  is  covered? 

4.  Why  is  the  pitch  of  a  fork  lowered  by  loading  its  prongs? 

5.  What  use  can  be  made  of  beats  in  tuning  two  sounding  bodies  to  the 
same  pitch? 

300.  Musical  Intervals.  — The  in/erva/  between  any  two  notes 
is  measured  by  the  ratio  of  the  vibration  number  of  the  higher  to 
that  of  the  lower.  Two  notes  are  said  to  be  in  unison  when  they 
have  exactly  the  same  pitch.     The  interval  between  two  notes  in 


232  Sound 

unison  is  unity  (the  ratio  of  equal  numbers).  If  the  vibration 
number  of  one  note  is  twice  that  of  another  (interval  =  2),  the 
first  is  said  to  be  an  octave  above  the  second.  Other  intervals  are 
considered  in  Art.  302. 

301.  Harmony  and  Discord.  —  If  the  effect  of  two  or  more 
tones  when  sounded  together  is  pleasing  to  the  ear,  the  tones  are 
said  to  be  harmonious  or  consonant;  if  the  effect  is  unpleasant, 
they  are  said  to  be  discordant  or  dissonant. 

The  two  wires  of  a  sonometer  (Fig.  144)  produce  one  contin- 
uous sound  when  tuned    to  exact  unison ;   but,  when  the  pitch 


^^Wy  Fh;  144. 

of  one  of  the  wires  is  gradually  raised  (either  by  shortening  the 
length  of  the  vibrating  portion  with  the  movable  bridge  or  by 
increasing  the  tension),  beats  are  heard.  The  beats  increase  in 
frequency  as  the  interval  is  increased ;  and,  as  they  become  too 
rapid  to  be  recognized  individually,  they  pass  from  an  unsteady, 
rattling  sound  into  a  discord.  As  the  interval  is  still  further 
increased,  the  tones  presently  become  less  discordant,  then  har- 
monious, then  again  discordant,  etc.  {Exp), 

Four  intervals  can  thus  be  found  between  the  original  tone  and 
its  octave  for  which  the  tones  are  in  harmony  ;  at  all  other  inter- 
vals between  these  limits  they  are  more  or  less  discordant.  It 
can  be  shown  that  these  four  intervals  are  {,  J,  f ,  and  |  respec- 
tively. These  ratios  are  frequently  called  simple  ratios  because 
they  are  expressible  in  small  numbers.  The  most  perfect  harmony 
is  that  of  a  tone  and  its  octave,  and  they  are  separated  by  the 
simplest  possible  interval  (|). 

302.  The  Major  Diatonic  Scale.  —  Musical  scales  are  deter- 
mined by  the  comparatively  few  intervals  that  are  pleasing  to  the 


Properties  of  Musical  Sounds  233 

ear.  The  major  diatonic  scale  consists  of  a  series  of  eight  notes 
whose  syllable  names  are 

do     re     mi   fa     sol    la      si     do^ 
The  intervals  between  the  first  or  keynote  and  the  other  notes 
of  the  scale  are  as  follows  :  — 

I    I     *     I     f    4    ¥    ^ 

These  numbers  are  sometimes  called  the  vibration  ratios  of  the 
notes. 
The  smallest  whole  numbers  expressing  the  same  ratios  are 

24    27     30     32     36    40     45     48 
The  intervals  between  successive  notes  of  the  scale  are 
do     re     mi   fa     sol    la     si    do 2 

I   ¥   if    I    ¥    S  H 

The  intervals  f  and  y^  are  called  tones ;  the  interval  |f  is  called 
a  semitone. 

The  eighth  note,  which  is  an  octave  above  the  first  and  is 
called  by  the  same  name,  is  taken  as  the  first  of  another  series  of 
eight  notes,  each  of  which  is  an  octave  above  the  note  of  the  same 
name  in  the  preceding  series.  The  scale  may  thus  be  repeated 
both  upward  and  downwarcl  over  as  many  octaves  as  is  desired. 

The  intervals  remain  the  same  whatever  the  absolute  pitch  of 
the  keynote  may  be.  When  middle  C  is  taken  as  the  keynote, 
the  letter  names  and  the  vibration  numbers  of  the  notes  of  the 
scale  are  as  given  below :  — 


f                                  , 

Position  on  the  stafi, 

1         1 

J           -'         ^ 

SI                                   _.           /^ 

€i 

•■^         1 

7       \              \            ^\            0            ^                                         1 

S--<sU     ^ 

Letter  names,                     c*        //'        ^'         /'         ^          a' 

^'         r" 

Syllable  names,                 do       re       mi       fa        sol        la 

si       do 

Vibration  numbers,          256     288     320     341J     384     426J 

480    5 1 2 

Vibration  ratios,                  ^          \         \           \           \           % 

¥     2 

Intervals  between 

successive  notes. 

f     ¥    il      t 

¥     1 

U 

234  Sound 

The  notes  do^  mi^  soly  do^  are  in  harmony,  and  constitute  the 
common  or  major  chord.  Their  vibration  numbers  are  propor- 
tional to  the  numbers  4,  5,  6,  8. 

303.  The  Tempered  Scale.  —  In  order  that  any  note  of  the 
diatonic  scale  may  be  taken  as  the  keynote,  it  is  necessary  to 
introduce  other  notes  within  the  octave.  These  notes  are  called 
sharps  or  flats,  and  are  played  on  the  piano  and  the  organ  by 
means  of  the  black  keys.  Five  notes  are  thus  added  in  the 
octave,  forming  the  chromatic  scale  — 

C    Cj    D     Djl    E    F    Fjf    G    G#    A    A#    B    C 

It  is  further  necessary  to  change  the  intervals  slightly,  making 
them  exactly  equal,  to  avoid  the  necessity  of  introducing  many 
more  notes.  Thus  the  intervals  C  to  D  and  D  to  E  are  equal, 
and  each  is  the  square  of  the  interval  E  to  F.  Only  a  trained 
ear  can  recognize  the  slight  imperfection  of  harmony  that  results 
from  this  equalization  of  the  intervals.  The  diatonic  scale  thus 
modified  is  called  the  tempered  scale  to  distinguish  it  from  the 
natural  scale  considered  in  the  preceding  article.  Only  the 
natural  scale  is  used  in  physics. 

304.  Limits  of  Audibility:  Range  of  Pitch  used  in  Music. — 
We  have  seen  that  the  vibrations  of  a  body  may  be  too  slow  to 
produce  an  audible  sound  (Art.  278).  The  sound  waves  from 
such  bodies  are  too  long  to  affect  the  ear.  In  all  other  respects 
they  are  like  the  waves  that  produce  the  sensation  of  sound.  The 
slowest  rate  of  vibration  that  will  produce  an  audible  sound  differs 
with  different  persons  whose  hearing  for  sounds  of  ordinary  pitch 
may  be  equally  good.  The  limit  of  audibility  generally  lies  be- 
tween 16  and  30  vibrations  per  second.  The  vibration  number 
of  the  lowest  note  of  a  piano  (A  of  the  fourth  octave  below  mid- 
dle C)  is  26.6.  The  lowest  note  of  most  pipe  organs  is  the  third 
C  below  middle  C,  which  gives  32  vibrations  per  second.^ 

1  It  is  an  interesting  fact  that  the  muscles  vibrate  when  in  action,  sounding  a  note 
near  the  lower  limit  of  audibility.  This  note  can  be  distinctly  heard  as  a  rapid  pul- 
sation by  pressing  the  palms  of  the  hands  firmly  over  both  ears.  The  sound  comes 
from  the  muscles  of  the  arms,  which  are  then  contracted. 


Properties  of  Musical  Sounds  235 

The  vibrations  of  a  body  may  also  be  too  rapid  to  produce  an 
audible  sound.  The  upper  limit  of  audibility  differs  much  more 
widely  with  different  persons  than  the  lower  Hmit.  From  numerous 
experiments  it  is  concluded  that  the  limit  generally  lies  between 
12,000  and  30,000  vibrations  per  second.  The  relative  Hmits  of 
audibility  of  the  members  of  the  class  can 

be  tested  by  gradually  raising  the  pitch  fTt&°tf[f^||)  '  fr"  ^  J 
of  a  Galton's  whistle  (Fig.  145)  {Exp.). 
It  is  said  that  some  persons  whose  hear- 
ing is  good  cannot  hear  the  shrill  chirping  of  a  cricket.  The 
highest  note  of  a  piano  is  the  fourth  C  above  middle  C  ;  its 
vibration  number  is  256  X  2^  or  4096.  Pitch  is  not  easily  or 
definitely  appreciated  beyond  this  limit. 

Laboratory  Exercise  46. 

305.  Conditions  affecting  the  Vibration  Rate  of  Strings.  —  Al- 
though the  sound  of  a  stringed  instrument  comes  almost  wholly 
from  the  body  of  the  instrument,  or  from  some  part  of  the  body, 
the  pitch  of  the  sounds  is  determined  by  the  rate  of  vibration  of 
the  strings.  (The  word  string  is  here  used  to  include  wires  as  well 
as  catgut  strings.)  It  is  well  known,  in  a  general  way,  that  the  pitch 
of  a  string  is  raised  by  shortening  it  {i.e.  the  vibrating  portion  of 
it),  or  by  increasing  its  tension ;  and  that  the  pitch  of  a  lighter 
string  is  higher  than  that  of  a  heavier  one  of  the  same  length 
when  under  the  same  tension.  The  effect  of  length  is  illustrated 
by  the  different  notes  obtained  from  the  same  string  of  a  violin, 
mandolin,  or  guitar,  by  varying  its  length  with  the  finger;  the 
effect  of  tension,  by  the  use  of  the  tightening  pegs  in  tuning  the 
strings ;  and  the  effect  of  mass,  by  the  different  strings  of  the  in- 
strument, the  heaviest  giving  the  lowest  note. 

The  effects  of  the  length,  tension,  and  mass  of  a  string  can  be 
illustrated  experimentally  by  means  of  a  sonometer  {Exp.),  They 
are  explained  in  a  general  way  {i.e.  qualitatively,  not  quantita- 
tively) as  follows  :  — 

306.  Discussion  of  the  Conditions.  —  Length.  —  It  will  be  re- 
membered that  when  a  pendulum  is  shortened,  it  vibrates  more 


236 


Sound 


rapidly  because  the  accelerating  force  upon  the  bob  for  a  given 
length  of  arc  is  increased  (Art.  135).  Similarly,  when  a  string  is 
shortened  without  change  of  tension,  the  accelerating  force  for  a 
given  displacement  (amplitude)  is  increased.  This  is  illustrated 
by  the  familiar  fact  that,  with  a  constant  tension,  the  shorter  a 
cord  is,  the  greater  is  the  resistance  that  it  offers  when  its  center  is 
pulled  a  given  distance  to  one  side.  But  the  shorter  cord  vibrates 
more  rapidly  for  a  second  reason  :  the  mass  to  be  carried  to  and 
fro  diminishes  in  the  same  ratio  as  the  length. 

Tension.  —  The  accelerating  force  for  a  given  amplitude  is  pro- 
portional to  the  tension ;  hence  an  increase  of  tension  increases 
the  rate  of  vibration. 

Mass.  —  'ITie  acceleration  produced  by  a  given  force  is  inversely 
proportional  to  the  mass  upon  which  the  force  acts  (Art.  113); 
hence,  other  conditions  being  equal,  the  greater  the  mass  of  a 
string,  the  slower  will  be  its  rate.  The  mass  of  a  string  per  unit 
length  is  proportional  to  its  density  and  also  to  the  square  of  its 
diameter. 

307.  The  Laws  of  Vibration  of  Strings. — The  following  laws 
of  vibration  of  strings  have  been  established  by  experiment ;  they 
are  derived  in  advanced  physics  by  mathematical  reasoning  based 
on  the  second  law  of  motion. 

I.  Other  conditions  remaining  constant,  the  vibration  number 
of  a  string  is  inversely  proportional  to  its  length. 

II.  Other  conditions  remaining  constant,  the  vibration  number 
of  a  string  is  directly  proportional  to  the  square  root  of  its  tension. 

III.  Other  conditions  remaining  constant,  the  vibration  number 
of  a  string  is  inversely  proportional  to  the  square  root  of  its  mass. 

The  first  of  these  laws  is  the  only  one  of  importance  in  elemen- 
tary physics.     We  find  use  for  it  in  the  following  article. 

PROBLEMS 

I.  Compute  the  wave  length  of  the  following  notes  in  air  at  20° : 
Ci  (  =  32),  C  (  =  64),  ^  (  =  4096),  and  the  highest  audible  sound,  assum- 
ing it  to  be  30,000. 


Properties  of  Musical  Sounds  237 

2.  The  pitch  of  a  string  50  cm.  long  is  r  (  =  128).  Compute  its  length 
for  each  note  of  the  octave,  the  tension  remaining  constant. 

3.  A  string  i  m.  long  sounds  C  (  =  64)  under  a  certain  tension.  The 
tension  remaining  constant,  compute  the  vibration  number  of  the  note  sounded 
by  2>  h  i»  ^»  6>  h  *"<!  i  o^  '^s  total  length.  All  these  notes  but  one  are  notes 
of  the  diatonic  scale  in  the  first  three  octaves  above  the  note  of  the  whole 
wire.     Identify  them. 

308.  Fundamental  Tone  and  Overtones.  —  By  properly  timing 
the  motion  of  the  hand  in  which  one  end  of  a  long  rubber  tube  or 
coil  of  wire  is  held,  the  tube  (or  wire)  can  be  made  to  vibrate 
regularly  as   a  whole  (Fig.   146).     When    the  frequency  of   the 


Fig.  147. 


impulses  is  doubled,  the  tube  vibrates  in  two  equal  segments  (Fig. 
147) ;  when  trebled,  it  vibrates  in  thirds.  The  segments  are 
separated  by  points  called  nodes,  which  remain  approximately 
at  rest  {Exp.). 

The  strings  of  a  musical  instrument  can  also  be  made  to  vibrate 
in  two  or  more  equal  segments.  To  illustrate  :  When  the  string  of 
a  sonometer  wire  is  bowed  near  one  end  and,  at  the  same  time,  is 
lightly  touched  at  its  mid  point  with  the  finger  nail  or  the  tip  of  the 
finger,  it  vibrates  in  halves,  sounding  the  octave  above  its  usual 
note,  even  after  the  bow  and  the  finger  are  removed  (the  finger 
being  removed  an  instant  after  the  bow).  When  the  string  is 
touched  at  one  third  its  length  from  one  end  and  bowed  or  plucked 
near  that  end,  it  vibrates  in  thirds  (Fig.  148).  In  the  same  man- 
ner the  string  can  easily  be  made  to  vibrate  in  any  number  of 


238  Sound 

equal  segments  up  to  eighths  or  tenths.    The  division  of  the  string 
into  segments  is  shown  at  a  distance  by  placing  upon  it  small 


Fig.  Z48. 

folded  bits  of  paper.    These  "  riders  "  are  thrown  off  by  the  vibra- 
tion except  at  or  very  near  the  nodes  (Ex^.). 

The  pitch  of  a  string  when  vibrating  in  segments  is  the  same  as 
it  would  be  if  the  string  consisted  of  but  one  of  the  segments. 
The  note  that  is  sounded  by  a  string  when  it  is  vibrating  as  a 
whole  is  called  its  fundamental  tone.  A  note  that  is  produced  by 
vibration  in  segments  is  called  an  ot^ertone  or  a  harmonic.  The 
overtones  are  numbered  in  order,  beginning  with  the  lowest :  thus 
the  first  overtone  is  produced  by  vibration  in  halves,  the  second 
by  vibration  in  thirds,  etc.  If  we  call  the  fundamental  tone  doi^ 
the  series  of  overtones  is  as  follows  :  — 

No.  of  segments,  I 

Names  of  the  overtones. 
Relative  pitch,  do^ 

309.  Quality  of  Sound.  —  The  quality  (Art.  294)  of  the  tone  of 
a  sonometer  wire  depends  largely  upon  whether  the  wire  is  plucked 
with  the  finger  or  the  finger  nail,  bowed,  or  struck  with  a  rubber 
mallet.  It  also  varies  with  the  place  where  the  wire  is  struck, 
bowed,  or  plucked  {Exp.). 

These  differences  in  quality  depend  upon  the  character  of  the 
vibration  of  the  string.  Probably  the  vibration  does  not  in  any 
case  consist  of  a  simple  to-and-fro  movement  of  the  string  as  a 


2 

3 

4 

5         6 

7 

8 

1st 

2d 

3ci 

4th     5th 

6th 

7th 

do^ 

soli 

do^ 

w/3    W3 

— 

do^ 

Properties  of  Musical  Sounds  239 

whole.  On  the  contrary,  it  is  generally  very  complex,  —  vibration 
in  halves,  thirds,  fourths,  etc.,  occurring  simultaneously  with  the 
vibration  as  a  whole.  Thus  the  note  consists  of  the  fundamental 
tone  together  with  one  or  more  (generally  several)  overtones. 
Figure  149  shows  different  forms  assumed  by  a  string  while  sound- 


FIG.  149. 

ing  its  fundamental  and  first  overtone.  The  dotted  lines  indicate 
the  forms  which  the  string  would  assume  if  it  were  sounding  only 
its  fundamental  tone.  In  some  cases  the  first  overtone  can  be 
separately  distinguished  when  an  effort  is  made  to  do  so  ;  but 
generally  the  overtones  are  not  recognized  as  separate  sounds, 
their  effect  being  to  impart  a  certain  quality  to  the  note.  When 
the  wire  is  struck  with  a  rubber  mallet  or  plucked  with  the  finger 
near  its  center,  the  fundamental  is  loud  and  all  the  overtones 
weak  ;  when  it  is  bowed  near  one  end,  the  overtones  are  much 
stronger,  adding  what  is  commonly  called  richness  to  the  sound. 
When  the  wire  is  plucked  near  one  end  with  the  finger  nail,  the 
fundamental  and  the  lower  overtones  are  wanting,  or  at  most  very 
weak,  and  the  higher  overtones,  some  of  which  are  discordant, 
produce  a  shrill  jangle. 


240 


Sound 


When  a  tuning  fork  is  struck,  it  sounds  one  or  more  shrill  over- 
tones, which  are  perfectly  distinct  from  the  fundamental.  These 
overtones  quickly  die  away,  leaving  the  fundamental  as  a  simple 
tone  {Ex/>.), 

It  was  proved  by  Helmholtz,  a  noted  German  physicist,  that 
the  quality  of  any  sound  is  determined  by  the  overtones  which 
accompany  its  fumlamental  tone.  Differences  in  the  quality  of 
sounds  are  due  to  the  presence  of  different  overtones  or  to  differ- 
ences in  their  relative  intensity.  The  tuning  fork  is  the  only  in- 
strument that  gives  a  simple  tone  ;  but  the  notes  of  the  diapason 
pipes  of  an  organ  are  quite  similar,  being  almost  free  from  over- 
tones. A  musical  sound  in  which  a  number  of.  the  lower  over- 
tones are  present  is  full  and  rich.  The  tones  of  the  piano  and 
the  violin  are  examples.  The  penetrating  character  of  the  tones 
of  brass  instruments  is  due  to  their  particularly  strong  overtones. 
'ITie  pitch  of  a  sound  is  the  pitch  of  its  fundamental  ;  it  is  not 
altered  by  the  accompanying  overtones. 

III.  Sympathetic  and  Forced  Vibrations:  Resonance 


Laboratory  Exercise  43. 
310.   Sympathetic  and  Forced  Vibrations. 
actions 


1 


Certain  mechanical 
which  occur  with 
sounding  bodies  may  be  illus- 
trated on  a  visible  scale  by 
means  of  the  apparatus  repre- 
sented in  Fig.  15b.  Two 
pairs  of  pendulums  are  sus- 
pended from  a  light  rod,  CZ>, 
the  pendulums  of  each  pair 
being  of  equal  length.  The 
rod  is  supported  by  short 
cords  from  a  fixed  support. 
^*°'  '^-  When  any  one  of  the   pen- 

dulums is  set  in  vibration,  it  pulls  the  rod  from  which  it  is  sus- 
pended back  and  forth. 


Sympathetic  and  Forced  Vibrations      241 

This  vibratory  motion  of  the  rod  imparts  a  series  of  impulses 
to  the  other  pendulums.  AUhough  the  effect  of  a  single  impulse 
is  almost  inappreciable,  a  succession  of  such  impulses  causes  the 
other  pendulum  of  the  same  length  to  vibrate  with  increasing 
amplitude ;  while  the  amplitude  of  the  first  steadily  decreases  as 
it  imparts  its  energy  to  the  other.  Since  the  two  pendulums  have 
the  same  rate  of  vibration,  the  impulses  imparted  to  the  one  that 
was  at  rest  are  rightly  timed  to  produce  a  cumulative  effect.  The 
vibrations  of  this  penduhmi  are  called  sympathetic^  signifying  that 
its  natural  rate  is  in  agreement  with  that  of  the  impulses  to  which 
it  responds. 

The  other  pair  of  pendulums  make  a  few  vibrations  with  in- 
creasing amplitude,  followed  by  an  equal  number  during  which 
the  amplitude  decreases  till  the  pendulums  are  brought  to  rest. 
This  series  of  increasing  and  decreasing  vibrations  is  repeated  in- 
definitely. The  explanation  of  this  behavior  is  that  the  impulses 
are  not  timed  in  agreement  with  the  rate  of  these  pendulums ;  a 
few  successive  impulses  produce  a  cumulative  effect,  but  these  are 
followed  by  an  equal  number  which  are  opposed  to  the  motion 
already  produced  and  hence  destroy  it.  Since  these  pendulums 
cannot  be  forced  to  vibrate  in  unison  with  the  impulses,  they 
cannot  accumulate  any  considerable  store  of  energy. 

When  the  rod  is  struck  or  drawn  to  one  side  and  released  (all 
of  the  pendulums  being  at  rest),  it  vibrates  much  more  rapidly 
than  it  does  when  under  the  control  of  a  vibrating  pendulum. 
Thus,  although  the  rod  has  a  natural  rate  of  vibration,  it  does  not 
persist  in  it  as  the  pendulums  do,  but  yields  to  the  impulses  of 
either  a  long  or  a  short  pendulum.  The  motion  of  the  rod  when 
thus  controlled  is  called  a  forced  vibration. 

311.  Forced  Vibrations  of  Sounding  Bodies.  —  The  sound  that 
comes  from  a  table,  an  empty  box,  or  a  large  board  when  touched 
by  a  vibrating  fork  is  due  to  the  forced  vibration  of  the  wood. 
Since  the  sound  thus  produced  always  has  the  same  pitch  as  the 
fork,  whatever  this  may  be,  it  is  evident  that  wood,  especially  in 
the  form  of  a  thin  board,  is  easily  forced  to  vibrate  in  unison  with 


242  Sound 

periodic  impulses  of  any  frequency  —  a  behavior  like  that  of  the 
rod  from  which  the  pendulums  were  suspended  (Fig.  150). 

When  a  violin  is  played,  the  sound  is  produced  by  the  forced 
vibration  of  the  body  of  the  instrument.  The  pitch  is .  deter- 
mined by  the  vibration  rate  of  the  strings ;  the  quality,  partly  by 
the  manner  in  which  the  string  is  bowed  and  partly  by  the  kind 
of  wood  of  which  the  instrument  is  made  and  the  workmanship. 
The  music  of  a  piano  comes  from  the  large  sounding  board  upon 
which  the  wires  are  strung. 

312.  Sympathetic  Vibration  of  Tuning  Forks  and  Strings.  —  A 
tuning  fork  may  be  made  to  vibrate  sympathetically  as  follows : 
The  stems  of  a  sounding  and  a  silent  fork  of  exactly  the  same 
pitch  are  touched  to  the  top  of  a  table  a  short  distance  apart. 
After  one  or  two  seconds,  the  fork  that  was  sounded  is  stopped 
with  the  fingers  or  removed  from  the  table,  and  a  sound  is  then 
heard  from  the  other  fork.  If  the  sound  is  too  faint  to  be  heard 
at  a  distance,  the  vibration  of  the  fork  can  be  proved  by  touching 
a  suspended  pith  ball  to  one  of  its  prongs.  This  vibration  is 
caused  principally  by  impulses  imparted  through  the  stem  by  the 
vibrating  table.  Probably  from  500  to  1000  such  impulses  are 
required  to  produce  the  observed  effect  {Exp.). 

The  silent  fork  can  also  be  made  to  vibrate  sympathetically  by 
holding  it  in  the  fingers  close  to  the  vibrating  fork,  held  in  the 
other  hand,  the  forks  facing  each  other  but  not  touching.  In 
this  case  the  impulses  are  imparted  by  the  sound  waves  in  the 
air  {Exp.). 

These  experiments  fail  if  the  forks  are  not  in  perfect  unison, 
as  will  be  found  by  repeating  them  with  one  of  the  forks  loaded 
with  wax  {Exp.). 

If  the  two  wires  of  a  sonometer  are  tuned  to  exact  unison, 
either  will  vibrate  sympathetically  when  the  other  is  sounded,  the 
impulses  being  transmitted  principally  through  the  body  of  the 
instrument.  As  the  pitch  of  one  of  the  wires  is  slowly  changed, 
the  response  of  the  silent  wire  immediately  becomes  very  faint 
and  quickly  ceases  {Exp,). 


Sympathetic  and  Forced  Vibrations      243 

These  experiments  show  that  tuning  forks  and  strings,  like  the 
pendulums,  persist  in  their  natural  rate  of  vibration,  —  a  behavior 
very  different  from  that  of  sounding  boards  and  the  bodies  of 
stringed  instruments. 

Laboratory  Exercise  44, 

313.  Resonance  of  Air  Columns.  —  The  reenforcement  of  the 
sound  of  one  body  by  the  sympathetic  or  forced  vibration  of 
another  is  called  resonance;  but  the  word  most  frequently  refers 
to  the  reenforcement  of  sound  by  the  sympathetic  vibration  of 
partly  inclosed  bodies  of  air.  We  shall  use  the  word  only  in  this 
narrower  sense. 

Resonance  may  be  secured  by  means  of  a  tube  provided  with  a 
close-fitting  piston  (Fig  151).     When  the  piston  is  in  a  certain 


£ 


Fig.  151. 

position,  the  sound  of  a  fork  held  at  the  end  of  the  tube  is  strongly 
reenforced  by  the  sympathetic  vibration  of  the  column  of  air 
extending  to  the  piston.  The  sound  from  the  tube  is  very  much 
fainter  when  the  piston  is  moved  even  a  very  short  distance  in 
either  direction,  from  the  position  of  maximum  reenforcement 
{Exp^.  The  column  of  air  will  vibrate  strongly  only  at  its  natural 
rate,  which  varies  with  the  length.  This  is  further  shown  by  repeat- 
ing the  experiment  with  a  fork  of  different  pitch.  It  will  be  found 
that  the  length  of  the  column  for  maximum  resonance  is  inversely 
proportional  to  the  vibration  number  of  the  note  (^Exp.). 

The  vibration  of  the  air  column  is  longitudinal.  As  the  nearer 
prong  of  the  fork  moves  toward  the  end  of  the  tube,  the  air  par- 
ticles in  the  tube  are  driven  a  short  distance  toward  the  farther 
end,  producing  a  condensation  which  is  greatest  at  the  piston. 
When  the  prong  moves  away  from  the  tube,  the  air  expands  by 
a  motion  of  the  particles  toward  the  mouth  of  the  tube.     The 


244 


Sound 


expansion,  like  the  compression,  is  greatest  at  the  piston.  Thus 
the  change  of  density  of  the  air  is  greatest  at  the  piston  and  di- 
minishes toward  the  open  end,  where  the  density  remains  nearly 
constant.  The  amplitude  of  vibration  of  the  air  particles  is 
greatest  at  the  open  end  and  diminishes  to  zero  at  the  piston. 
The  tube  controls  the  vibration  of  the  air  for  a  short  distance 
beyond  its  end,  and  it  has  been  found  by  experiment  that  the  true 
length  of  the  air  column  is  equal  to  the  length  of  the  tube  (to  the 
piston)  plus  half  the  diameter  of  the  tube.  It  can  be  shown  that 
this  length  is  one  fourth  of  a  wave  length,  i.e.  is  one  fourth  the 
length  of  the  sound  wave  set  up  by  the  vibration  of  the  fork  and 
the  air  column. 

Second  resonance  occurs  when  the  length  of  the  air  column  is 
increased  to  three  fourths  of  a  wave  length.  The  mode  of  vibra- 
tion of  the  air  is  shown  in  Fig.  152,  which  represents  the  condi- 


Y 


||!|llllllillllllllillll!llli!| 


ii:: uimwimii 


liiiiil 


m 
■ 


I 


\  I  / 


ami 


Fi<; 


tion  of  the  air  at  intervals  of  one  fourth  of  a  vibration.  AB  is 
the  length  for  first  resonance,  AD  for  second  resonance.  AB,  BC, 
CD,  are  each  one  fourth  of  a  wave  length.  At  B  and  D  there  is 
no  motion,  and   the  change  of  density  is  the   greatest.    These 


Sympathetic  and   Forced  Vibrations     245 

positions  are  called  nodes.  (Compare  with  the  nodes  of  a  string 
vibrating  in  segments.)  At  A  and  C  there  is  little  or  no  change 
of  density  and  the  amplitude  of  vibration  is  greatest.  These 
positions  are  called  atiiinodes.  The  arrows  in  the  second  and 
fourth  parts  of  the  figure  indicate  the  direction  of  motion  of  the 
air  particles.  At  the  instant  represented  in  the  first  and  third 
parts  of  the  figure  the  air  is  at  rest  throughout  the  tube.  BD 
is  a  vibrating  segment  of  the  air  column;  its  length  is  half  a 
wave  length.     AB  is  half  a  segment. 

If  the  tube  were  further  lengthened  by  half  a  wave  length, 
another  whole  segment  would  be  added  to  the  vibrating  air 
column  and  third  resonance  would  occur. 

314.  Resonance  caused  by  Noise.  —  Any  partly  inclosed  body 
of  air  has  a  natural  rate  of  vibration  as  a  whole,  and,  when  sub- 
jected to  a  series  of  impulses  rightly  timed,  it  will  vibrate  sympa- 
thetically. A  noise  always  affords  such  a  series  of  impulses,  since 
it  consists  of  an  indefinite  number  of  different  rates  of  vibration. 
Hence  a  hollow  body,  as  a  tumbler,  a  glass  or  metal  tube,  or  a 
sea  shell,  continually  sounds  a  faint  note,  which  is  distinctly  heard 
when  the  body  is  held  to  the  ear.  The  pitch  of  this  sound  re- 
mains constant  when  different  noises  are  made  in  the  vicinity, 
but  its  loudness  varies,  in  many  cases  swelling  to  a  loud  roar  when 
the  foot  is  scraped  on  the  floor  {Exp.). 

315.  The  Whistle.  —  In  wind  instruments  the  sounding  body 
is  a  column  of  air  confined  in  a  tube.  The  whistle  is  a  familiar 
illustration.  Its  action  is  explained  as  follows :  A  current  of  air 
passing  through  a  narrow  slit,  a  (Fig.  153),  is  directed  against 
the  farther  edge  of  a  lateral  opening,  b.  Con- 
tact with  this  edge  causes  the  current  of  air  to  3  '  r-—^ 
flutter   irregularly,  producing  a  faint  rustling 

noise.     The  column  of  air,  c,  confined  in  the 
body  of  the  whistle  is  thrown  into  strong  sympathetic  vibration  by 
those  impulses  from  the  current  of  air  which  are  in  unison  with  its 
natural  rate.    The  note  thus  produced  is  generally  so  much  louder 
than  the  noise  of  the  air  current  that  the  latter  is  not  heard.     By 


246 


Sound 


means  of  a  whistle  fitted  with  a  piston  it  can  be  shown  that  the 
pitch  of  the  sound  rises  as  the  size  of  the  air  chamber  is  de- 
creased {Exp.  with  Ga/ton's  whistle  or  organ  pipe  with  piston). 
Within  wide  Hmits  of  pitch,  the  inclosed  air  always  finds  its  own 
note  in  the  noise  of  the  air  current,  and  reenforces  it. 

316.  Organ  Pipes.  —  An  organ  pipe  is  merely  a  whistle  of  spe- 
cial construction.     The  pitch  of  its  note    is  determined  by  the 

length  of  the  air  column  (cor- 
rected for  the  diameter  of  the 
pipe).  The  quality  of  the 
sound  is  modified  by  the  shape 
and  material  of  the  pipe. 
Figure  154  represents  a  rec- 
tangular wooden  pipe  and 
Fig.  155  a  cylindrical  pipe 
of  metal.  Some  pipes  are 
provided  with  a  tongue  or 
reed,  against  which  the  cur- 
rent of  air  from  the  bellows 
is  directed,  causing  it  to  vi- 
brate. The  reed  is  tuned  to 
the  natural  rate  of  the  air 
column  in  the  pipe,  which 
therefore  vibrates  sympatheti- 
cally. The  note  of  a  reed 
Organ  pipes  are  made  both 

open  and  closed  at  the  top ;  the  latter  are  called  stopped  pipes. 

The  following  laws  may  be  illustrated  by  means  of  a  pipe  provided 

with  a  piston  :  — 

I.  The  vibration  number  of  an  air  column  is  inversely  propor- 
tional to  its  length. 

II.  The  pitch  of  an  open  pipe  is  an  octave  above  that  of  a  closed 
pipe  of  the  same  length. 

317.  Fundamental  Tone   and  Overtones  of   Organ  Pipes.  —  In 
sounding  its  fundamental  note  the  air  in  a  closed  pipe  vibrates 


Fig. 


Fig. 


155- 


pipe  has  a  characteristic  quality. 


or  THF 

UNIVERSI7 
Sympathetic  and  Forced  Vibrati\jsc;,,  /^^^v^ 

in  a  half  segment,  with  an  antinode  at  the  mouth  and  a  node  at 
the  closed  end,  as  a  resonance  tube  does  for  first  resonance  (Art. 
313).  Thus  the  length  of  a  stopped  pipe  is  one  fourth  the  wave 
length  of  its  fundamental  tone.  The  first  overtone  is  produced 
by  vibration  in  one  and  one  half  segments,  as  for  second  reso- 
nance (Art.  313).  Its  vibration  number  is  three  times  that  of 
the  fundamental,  and  hence  corresponds  to  the  second  overtone 
of  a  string. 

The  second  overtone  is  produced  by  vibration  in  two  and  one 
half  segments,  and  corresponds  to  the  fourth  overtone  of  a  string. 
When  a  pipe  is  sounded  by  a  gentle  current  of  air,  only  its  funda- 
mental is  heard ;  with  a  greater  pressure  of  air,  this  gives  place 
to  the  first  overtone ;  and,  with  still  greater  pressure,  the  second 
overtone  is  sounded  {Exp.). 

The  air  in  an  open  pipe  necessarily  vibrates  with  an  antinode  at 
each  end.  When  sounding  its  fundamental,  there  is  but  one  node, 
and  this  is  at  the  middle ;  i.e.  the  air  vibrates  in  two  half  segments. 
The  length  of  an  open  pipe  is  therefore  half  the  wave  length  of 
its  fundamental  tone.  The  first  overtone  is  produced  by  vibra- 
tion with  one  segment  between  the  half  segments  at  the  ends,  the 
second  overtone  with  two  segments  between,  the  third  with  three, 
etc.  From  the  relative  length  of  the  segments  thus  produced  it 
will  be  seen  that  the  complete  series  of  overtones  may  be  present 
as  with  strings.  The  difference  in  the  overtones  present  causes 
a  difference  in  the  quality  of  open  and  closed  pipes. 

318.  Wind  Instruments.  —  Each  pipe  of  an  organ  sounds  only  one 
note ;  hence  in  a  complete  set  of  pipes  there  is  one  for  each  note 


of  the  instrument.     An  organ   is  provided  with  several  sets  of 
pipes,  the  notes  of  each  set  differing  in  quality  from  those  of  the 


248 


Sound 


other  sets.     Most  wind  instruments  are  provided  with  but  one 
tube,  the  different  notes  being  produced  either  by  varying  the 

length  of  the  tube   or  by 
sounding  overtones. 

In  the  trombone  (Fig. 
156),  one  part  of  the  tube, 
SL^  sHdes  within  the  other, 
and  the  tube  is  shortened 
by  pushing  the  shding  part 
farther  in.  The  tube  of  the 
cornet  (Fig.  157)  forms  sev- 


FiG.  157. 


eral  turns  or  convolutions,  which  may  either  be  included  or  cut 
off  from  the  remainder  of  the  tube  by  means  of  pistons,  <z,  by  r, 
thus  varying  the  length  of  the  vibrating  column  of  air. 

The  bugle y  the  trumpet  (Fig.  158),  and  the  coaching  horn  are 
without  any  device  for  altering  the 
length  of  the  air  column  ;  hence  their 
notes  are  limited  to  the  fundamental 
and  the  complete  series  of  overtones. 
The  different  notes  are  produced  by  ^'^'-  ^58. 

varying  the  force  of  the  breath  and  the  tension  of  the  lips. 

The  flute  and  clarinet  are  provided  with  holes  at  different  dis- 
tances, which  are  closed  by  the  fingers  or  by  keys.  An  antinode 
is  produced  at  an  open  hole ;  this  modifies  the  division  of  the  air 
column  into  vibrating  segments,  and  hence  determines  the  note. 


PROBLEMS 

1.  Gjmpute  the  length  of  a  stopped  pipe  that  sounds  Ci  (=  32)  at  20°  C, 
making  no  allowance  for  the  diameter. 

2.  Compute  the  length  of  an  open  pipe  that  sounds  ^'  (  =  256) ;   also  the 
length  of  one  that  sounds  ^  (=  4096). 

3.  What  are  the  letter  and  syllable  names  of  the  first  five  overtones  of  an 
open  pipe  whose  fundamental  is  c'  or  do\  ? 

4.  What  are  the  letter  and  the  syllable  names  of  the  first  two  overtones 
of  a  closed  pipe  of  the  same  pitch  as  in  the  preceding  problem  ? 


The  Ear  and  the  Voice 


249 


IV.   The  Ear  and  the  Voice 

319.  The  Parts  of  the  Ear.  —  The  ear  consists  of  three  parts ; 
namely,  the  external  ear^  the  middle  ear  or  tympanum,  and  the 
ititernal  ear  or  labyrinth.  The  sensation  of  sound  originates  in 
the  internal  ear ;  the  other  parts  are  accessory. 

320.  The  External  Ear.  —  The  external  ear  consists  of  the 
expanded  part  on  the  exterior  of  the  head  and  the  passage  lead- 
ing inward  from  it.    This  passage  is  a  little  over  an  inch  in  length, 


and  is  closed  at  its  inner  end  by  a  thin  membranous  partition, 
called  the  tympanic  or  drum  membrane.  It  is  popularly  called  the 
eardrum ;  but  this  term  is  more  properly  applied  to  the  middle 
ear.  The  drum  membrane  is  kept  under  tension  by  a  small 
muscle,  which  pulls  it  in  at  the  center,  giving  it  a  conical  shape. 
Figure  159  represents  the  front  view  of  a  section  of  the  right  ear. 
The  expanded  part  of  the  external  ear  concentrates  the  sound 
waves  and  directs  them  into  the  passage,  at  the  inner  end  of 
which  they  beat  upon  the  drum  membrane,  setting  it  into  forced 
vibration. 


! 


250  Sound 

321.  The  Middle  Ear.  —  The  middle  ear,  or  tympanum,  P 
(Fig.  159),  is  an  irregular  cavity  in  the  temporal  bone,  separated 
from  the  external  ear  by  the  dniin  membrane.     An  enlarged  view 

of  this  cavity  is  shown  in  Fig.  160. 
The  tympanum  is  connected  with  the 
back  part  of  the  mouth  (pharynx)  by 
the  Eustachian  tube,  R  (Fig.  159). 
This  tube  allows  air  to  pass  between 
the  throat  and  the  middle  ear,  thus 
equalizing  the  pressure  on  the  two 
sides  of  the  drum  membrane. 

A  chain  of  three  small  bones  ex- 
tends across  the  middle  ear  from  the 
*°*  '  drum  membrane  to  the  internal  ear. 

They  are  called  from  their  shape,  the  hammer,  ;//,  the  anvil,  i,  and 
the  stirrup,  s.  A  long,  slender  portion  of  the  hammer  (the  handle) 
is  attached  to  the  inner  surface  of  the  drum  membrane  ;  its  enlarged 
end  or  head  is  fastened  to  an  end  of  the  anvil  by  an  almost  im- 
movable joint  The  other  end  of  the  anvil  is  jointed  to  the 
stirrup ;  and  the  foot  plate  of  the  latter  is  fastened  to  a  mem- 
brane only  slightly  larger  than  itself,  which  closes  an  oval  opening 
(the  avai  window)  in  the  bony  partition  between  the  middle  and 
the  internal  ear. 

In  vibrating,  the  drum  membrane  moves  inward  and  outward 
through  a  distance  that  is  probably  not  more  than  .1  mm.  at  the 
most,  and  in  this  motion  carries  the  handle  of  the  mallet  with  it. 
The  mallet  and  the  anvil  are  supported  by  attachments  not  shown 
in  the  figures,  which  permit  their  rotation  as  one  body  about  an 
axis  passing  through  the  head  of  the  mallet  and  perpendicular  to 
the  plane  of  the  paper  in  the  figures.  Hence,  as  the  two  bones 
rotate  together,  each  inward  and  outward  movement  of  the  handle 
of  the  mallet  causes  a  like  movement  of  the  stirrup.  Thus  the 
vibrations  of  the  drum  membrane  are  transmitted  by  means  of  the 
chain  of  bones  to  the  membrane  covering  the  oval  window,  and, 
by  the  latter,  to  the  liquid  in  the  internal  ear. 


The  Ear  and  the  Voice 


251 


322.  The  Internal  Ear.  —  The  internal  ear,  or  labyrinth^  occu- 
pies several  chambers  and  tubes  hollowed  out  in  the  temporal 
bone.  The  middle  chamber,  called  the  vestibule^  F(Fig.  159),  is 
adjacent  to  the  inner  side  of  the  middle  ear.  Behind,  the  vesti- 
bule opens  into  three  semicircular  canals,  one  of  which  is  shown 
at  b  in  the  figure ;  in  front,  it  opens  into  a  spirally  coiled  tube,  5, 
called  the  cochlea,  from  its  resemblance  to  the  shell  of  a  snail.  In 
these  bony  chambers  and  tubes  lie  membranous  chambers  and 
tubes,  in  certain  parts  of  which  the  fibers  of  the  auditory  nerve.  A, 
end.  The  space  both  within  and  without  the  membranous  parts 
is  filled  with  a  watery  liquid.  The  function  of  the  semicircular 
canals  seems  to  be  distinct  from  that  of  hearing. 

323.  The  Cochlea.  —  The  bony  tube  of  the  cochlea  is  coiled 
into  a  spiral  of  two  and  a  half  turns.  It  contains  a  spiral  ledge  of 
bone,  which  projects  into  it  from  the  axis  of  the  spiral  and  follows 
the  winding  of  the  tube  throughout  its  length  (Fig.  159).  A  cross- 
section  of  the  bony  tube  of  the  cochlea  is  represented  diagram- 
matically  in  Fig.  161  ;  in  which  Sp  is  the  spiral  ledge,  and  C  a 
triangular,  membranous  tube,  called  the  canal  of  the  cochlea  (not 
shown  in  Fig.  159),  which  is  attached  at  its  inner  edge  to  the 
spiral  ledge  and  at  its  outer  surface  to  the  wall  of  the  tube.  The 
space  outside  the  canal  is  thus  divided  into  an  upper  channel,  A^ 
and  a  lower  channel,  B.  These  channels  unite  at  the  tip  of  the 
cochlea  around  the  end  of  the  canal,  which  is  closed.  The  lower 
end  of  the  upper  channel  is  connected  with 
the  vestibule  ;  the  lower  channel  terminates 
at  a  round  opening,  r  (Fig.  159),  in  the 
bony  partition  between  the  internal  and 
the  middle  ear.  This  opening  is  called 
the  round  window.  It  lies  just  below  the 
oval  window,  and,  like  the  latter,  is  covered 
with  a  membrane.  The  channels  and  the 
canal  are  filled  with  a  watery  liquid.  '°'  '^^* 

The  lower  side  of  the  canal,  vib  (Fig.  161),  is  called  the  basilar 
(basal)    membrane.     It  consists  of  thousands  of  delicate  fibers 


252  Sound 

extending  across  the  canal  from  side  to  side.  This  membrane 
possesses  upon  its  inner  face,  along  the  whole  length  of  the  tube, 
from  the  top  to  the  bottom  of  the  spiral,  a  very  remarkable  cellular 
structure  known  as  the  or-gan  of  Corti,  The  filaments  of  the  audi- 
tory nerve  terminate  among  the  cells  of  this  structure. 

The  vibration  of  the  membrane  to  which  the  stirrup  is  attached 
causes  a  vibration  of  the  liquid  in  the  vestibule.  This  vibration  is 
transmitted  along  the  upper  channel  of  the  tube  of  the  cochlea  to 
its  upper  end;  thence  back  along  the  lower  channel  to  the  round 
window.  The  yielding  of  the  membrane  that  covers  this  opening 
permits  vibrations  of  much  larger  amplitude  in  the  cochlea  than 
would  otherwise  be  possible.  These  vibrations  are  imparted  to 
the  membranous  canal  of  the  cochlea  and  the  liquid  within  it. 
Now  the  lower  side  of  the  canal  (the  basilar  membrane)  gradually 
widens  from  its  lower  to  its  upper  end  as 
shown  diagram matically  in  Fig.  162  ;  and 
is  supposed  to  vibrate  sympathetically  in 
transverse  segments  (as  <x,  b,  and  c  in  the 
figure),  the  rate  of  each  segment  depending  in  part  upon  its  length. 
Thus  when  a  number  of  vibrations  of  different  frequencies,  such  as 
constitute  an  ordinary  musical  sound,  are  transmitted  to  the 
cochlea,  they  (it  is  supposed)  throw  into  sympathetic  vibration 
those  parts  of  the  basilar  membrane  which  have  the  same  natural 
rate,  and  the  filaments  of  the  auditory  nerve  ending  in  the  over- 
lying parts  of  the  organ  of  Corti  are  stimulated.  According  to 
this  theory,  the  recognition  of  pitch  is  due  to  the  stimulation  of 
different  nerve  fibers  by  different  rates  of  vibration. 

324.  The  Larynx  and  the  Vocal  Bands.  —  At  the  upper  end 
of  the  windpipe  there  is  a  short  tubular  box,  called  the  larynx, 
which  opens  above  into  the  back  part  of  the  mouth,  just  below  the 
base  of  the  tongue.  Figure  1 63  represents  a  vertical  section  through 
the  larynx  with  the  hinder  half  removed.  The  larynx  is  made  of 
plates  of  cartilage,  movable  on  one  another,  and  connected 
together  by  joints,  muscles,  and  membranes. 

At  the  middle  of  the  larynx  there  is  a  projecting  ridge  on  each 


The  Ear  and  the  Voice 


253 


Fig.  163. 


side,  whose  edges  extend  from  front  to  back.  These  ridges  are 
composed  of  elastic  tissue,  and  are  cov- 
ered with  a  mucous  membrane  :  their  sharp 
free  edges  are  the  so-called  vocal  cords^  V 
(Fig.  163),  whose  vibration  produces  the 
voice.  They  are  more  appropriately  called 
vocal  bands.  The  large  angular  plate  of 
cartilage,  Thy  to  which  the  vocal  bands 
are  attached  in  front,  causes  the  promi- 
nence in  the  neck  known  as  "  Adam's 
apple." 

The  slitlike  opening  between  the  vocal 
bands  is  called  the  glottis.  In  quiet 
breathing  this  opening  is  narrow  in  front 
and  wider  behind,  leaving  a  free  passage 
for  the  air  to  and  from  the  lungs  as  shown 
in  Fig.  164,  By  which  represents  a  top  view  of  the  larynx  and  the 
vocal  bands.  In  using  the  voice,  the  bands  are  brought  close 
together  at  the  rear  by  muscular  action,  leaving  only  a  narrow  slit 


Epiglottis 
False  Vocal  Cords 


True  Vocal  Cords" 


Fig.  164. 

between  their  parallel  edges  (Fig.  164,  A).  When  the  bands  are 
in  this  position,  they  are  set  in  vibration  by  the  current  of  air  from 
the  lungs,  and  the  sound  thus  produced  constitutes  the  voice. 

325.  The  Voice.  — The  vocal  bands  alone  would  produce  only  a 
very  feeble  sound  ;  but  their  vibration  is  reenforced  and  modified 
in  quality  by  the  sympathetic  vibration  of  the  air  in  the  pharynx 
and  in  the  mouth  and  nasal  cavities,  which  together  constitute  a 
resonance  chamber.     The  loudness  of  the  voice,  aside  from  the 


254  Sound 

reenforcement  by  resonance,  depends  upon  the  force  with  which 
the  air  is  driven  past  the  bands,  together  with  the  size  and  condi- 
tion of  the  bands  themselves. 

The  pitch  of  the  voice  depends  primarily  on  the  size  of  the 
lar)'nx,  —  the  longer  the  vocal  bands  are,  the  lower  is  the  pitch. 
It  is  for  this  reason  that  the  voices  of  men  are  lower  in  pitch  than 
those  of  women  and  children.  Any  note  within  certain  limits  can 
by  produced  at  will  by  the  action  of  the  muscles  in  the  larynx, 
which  alter  the  tension  of  the  vocal  bands. 

The  quality  of  the  voice  depends  primarily  upon  the  natural 
size  and  form  of  the  larynx  and  the  different  resonance  cavities, 
and  the  condition  of  the  mucous  membrane  lining  these  cavities 
and  covering  the  vocal  bands.  But  the  quality  is  also  largely 
affected  by  the  manner  in  which  the  vibration  of  the  vocal  bands 
and  the  form  of  the  resonance  cavities  are  controlled  by  muscular 
action ;  and  it  may  therefore  be  modified  and  improved  by  culti- 
vation. 

326.  Speech.  —  The  modulation  of  the  voice  into  speech  is 
effected  by  changing  the  size  and  form  of  the  pharynx  and  the 
mouth  and  nasal  cavities  by  movements  of  the  soft  palate,  tongue, 
cheeks,  and  lips.  "  By  movements  of  tongue,  lips,  and  palate,  the 
air  current,  and  therefore  the  sound,  is  interrupted  from  time  to 
time  ;  on  other  occasions  the  air  is  forced  through  a  narrow  pas- 
sage in  the  mouth,  giving  rise  to  new  sounds  added  on  to  those 
originated  by  the  vocal  cords.  In  such  manner  the  primitive, 
feeble,  monotonous  tone  due  to  the  vocal  cords  is  reenforced  and 
altered  in  various  ways  in  throat  and  mouth,  and  voice  is  devel- 
oped into  articulate  speech^  —  Martin's  The  Human  Body, 


CHAPTER  X 

LIGHT 

I.   Nature  and  Transmission  of  Light 

327.  Ways  in  which  Energy  can  be  Transmitted.  —  There  are 
only  two  conceivable  ways  in  which  energy  can  be  transmitted. 
The  one  consists  in  the  transmission  of  matter  possessing  energy, 
the  other  in  the  transmission  of  a  disturbance  of  some  sort  through 
matter.  A  flying  bullet  and  a  moving  express  train  are  exam- 
ples of  the  first;  sound  waves  and  water  waves,  of  the  second. 
In  one  case  the  energy  remains  associated  with  the  same  body 
of  matter  during  transmission ;  in  the  other  it  is  handed  on  to  suc- 
cessive portions  of  the  medium. 

328.  Theories  of  Light :  Historical.  —  The  two  ways  in  which 
energy  can  be  transmitted  served  as  a  basis  for  two  rival  theories 
of  light,  which  were  supported  by  different  physicists  during  the 
latter  part  of  the  seventeenth  century.  They  are  known  as  the 
emission  or  corpuscular  theory,  and  the  undulaiory  or  wave  theory. 

According  to  the  emission  theory,  a  luminous  body  emits  streams 
of  minute  particles  (corpuscles),  which  travel  in  straight  lines,  and 
on  entering  the  eye,  cause  vision  by  their  impact  on  the  retina. 
According  to  the  wave  theory,  all  space  that  is  otherwise  unoccu- 
pied is  filled  with  a  substance  called  the  luminiferous  eiher^  or 
simply  the  ether;  and  light  consists  in  a  periodic  disturbance 
transmitted  through  this  medium  from  luminous  bodies,  in  much 
the  same  way  as  sound  is  transmitted  through  the  air  from  sound- 
ing bodies. 

Neither  theory,  as  at  first  advanced,  was  fully  satisfactory ;  but 
Newton's  support  of  the  emission  theory  carried  the  decision 
in  its  favor,  and  it  met  with  almost  universal  acceptance  during 

255 


256  Light 

the  eighteenth  centur>'.  At  the  beginning  of  the  nineteenth  century 
the  subject  was  again  taken  up  by  a  number  of  able  physicists. 
By  a  series  of  remarkable  experiments  adtiitional  facts  were 
brought  to  light,  which  amounted  to  conclusive  evidence  against 
the  emission  theory  and  in  favor  of  the  wave  theory.  The  latter, 
with  some  comparatively  recent  modifications,  is  now  regarded  as 
fully  established.  The  facts  and  experiments  that  led  to  the  final 
adoption  of  the  wave  theory  lie  almost  wholly  beyond  the  scope 
of  elementary  physics ;  we  must,  however,  make  use  of  the  sim- 
pler elements  of  the  theory  in  explaining  the  phenomena  with 
which  we  have  to  deal. 

329.  The  Ether.  —  The  ether  fills  all  space  throughout  the 
known  universe,  for  it  is  only  by  means  of  light  that  reaches  us 
from  the  heavenly  bodies  that  we  have  any  knowledge  of  them. 
It  cannot  be  excluded  from  a  vacuum ;  in  fact,  there  is  no  evi- 
dence that  the  quantity  of  it  in  a  receiver  is  diminished  in  the 
slightest  degree  by  whatever  means  the  receiver  may  be  exhausted. 
The  velocity  of  light  through  transparent  solids,  liquids,  and  gases 
is  enormously  greater  than  it  would  be  if  these  bodies  served  as 
media  by  means  of  which,  as  well  as  through  which,  the  light  is 
transmitted ;  hence  it  is  concluded  that  light  is  transmitted 
through  bodies  by  means  of  the  ether  which  fills  the  spaces 
between  their  molecules. 

While  the  ether  must  be  regarded  as  a  form  of  matter,  it  is  in 
all  probability  millions  of  times  less  dense  than  air  under  atmos- 
pheric pressure,  and  there  is  no  evidence  either  that  it  is  subject 
to  gravitation  or  that  it  is  composed  of  molecules.  On  the  con- 
trary, from  the  manner  in  which  it  transmits  disturbances,  it  is 
thought  to  be  perfectly  continuous  and  incompressible.  It  is 
supposed  to  fill  the  intermolecular  spaces  in  all  bodies  fi-om  the 
most  highly  rarefied  to  the  densest. 

330.  Nature  of  Radiation.  —  Some  helpful  ideas  in  regard  to 
the  origin  and  transmission  of  radiant  energy  (including  light) 
can  be  gathered  from  a  comparison  with  the  phenomena  of 
sound.    A  sounding  body,  as  we  have  learned,  is  the  center  of  a 


Nature  and  Transmission  of  Light     257 

periodic  disturbance  which  is  radiated  (transmitted  radially) 
from  it  in  all  directions  through  the  surrounding  medium, 
in  the  form  of  concentric  spherical  shells,  called  sound  waves. 
Now  the  molecules  of  all  bodies,  as  we  have  also  learned,  are  sup- 
posed to  be  in  constant  and  inconceivably  rapid  vibration  (Art. 
186).  According  to  the  wave  theory,  the  vibrations  of  each  mole- 
cule disturb  the  surrounding  ether;  and,  as  in  the  case  of  sound, 
this  disturbance  is  radiated  with  equal  velocity  in  all  directions 
as  a  train  of  concentric  spherical  waves.  Thus  each  molecule  of 
a  body  may  be  compared  to  a  sounding  fork,  and  the  entire  body 
to  a  large  group  of  forks,  each  of  which  is  sending  out  a  train  of 
waves.  As  the  energy  of  a  sounding  body  is  gradually  imparted 
to  the  air  or  other  medium,  so  too  the  heat  of  bodies  is  gradually 
imparted  to  the  ether  as  energy  of  wave  motion,  and  in  this  form 
it  is  called  radiant  energy. 

It  must  not  be  supposed,  however,  that  sound  waves  and  ether 
waves  are  at  all  alike  in  character ;  we  should  rather  expect  the 
contrary,  since  the  properties  of  the  ether  are  very  different  from 
those  of  ordinary  matter.  Ether  waves  consist  of  a  periodic  dis- 
turbance 0/  some  sort;  they  have  a  definite  length  (measured 
radially,  as  in  the  case  of  sound  waves)  and  travel  with  a  definite, 
though  very  great,  velocity ;  but  they  do  not  consist  of  conden- 
sations and  rarefactions  and  the  vibrations  are  not  longitudinal. 

331.  Luminous  and  Nonluminous  Radiation.  —  All  known  bodies 
are  a  constant  source  of  radiant  energy,  since  all  possess  some 
degree  of  heat ;  but  bodies  colder  than  their  surroundings  lose  less 
heat  than  they  receive  by  the  absorption  of  radiation  falling  upon 
them,  and  hence  become  warmer.  With  few  exceptions,  bodies 
emit  only  nonluminous. radiation  (the  so-called  "radiant  heat") 
unless  they  are  very  hot.  A  piece  of  iron,  for  example,  becomes 
luminous  at  about  525°,  at  which  temperature  it  emits  a  dull  red 
light.  As  the  temperature  increases,  the  light  grows  stronger  and 
changes  in  color,  at  last  becoming  white.  The  nonluminous  radi- 
ation also  increases  in  intensity  as  the  temperature  rises,  as  is  shown 
by  the  greater  heating  of  the  hand  when  held  near  the  body. 


258 


Light 


When  a  body  is  heated,  its  molecules  vibrate  more  rapidly  and 
hence  set  up  shorter  waves  in  the  ether, —  an  effect  similar  to  that 
produced  by  raising  the  pitch  of  a  sounding  body.  As  there  are 
sound  waves  too  long  and  others  too  short  to  cause  the  sensation 
of  sound,  so  too  there  are  ether  waves  both  too  long  and  too  short 
to  cause  sight.  Ether  waves  of  the  proper  length  to  stimulate 
the  optic  nerve  and  cause  the  sensation  of  vision  are  called  light. 
The  distinction  between  light  and  nonluminous  radiation  is  there- 
fore primarily  a  physiological  one  (Arts.  221  and  222). 

332.  The  Propagation  of  Light  in  a  Homogeneous  Medium.  — 
Any  space  or  substance  through  which  light  can  travel  is  called  a 
medium.  A  medium  is  said  to  be  homogeneous  when  its  chemical 
composition  and  density  are  the  same  in  all  parts  of  it.  Although 
light  is  transmitted  or,  as  we  commonly  say,  propagated  in  any 
medium  by  means  of  the  ether  in  its  intermolecular  spaces,  the 
substance  itself  affects  the  process  in  different  ways,  as  will  be  seen 
later. 

When  a  beam  of  sunlight  is  admitted  into  a  darkened  room, 
its  path,  which  is  rendered  visible  by  the  dust  particles  in  the  air, 
is  seen  to  be  perfectly  straight  {Exp.).  This  is  perhaps  the  best 
illustration  of  a  very  general  law ;  namely,  In  aiety  homogeneous 
medium  light  travels  in  straight  lines.  The  most  familiar  conse- 
quence of  this  fact  is  the 
formation  of  shadows.  The 
light  that  passes  on  either 
side  of  an  object  continues 
in  a  straight  line  ;  if  it  bent 
round  into  the  space  behind 
ttie  object,  there  would  be 
no  shadow. 

Figure  165   is  a  section 
diagram  in  which  each  circle 
represents  a  wave  of  light 
from  a  small  luminous  source 
at/.    The  light  that  passes  through  an  opening,  (9,  in  a  screen. 


Nature  and  Transmission  of  Light      259 


> 
> 
> 


m 


Fig.  166. 


AB,  is  represented  by  C/D.     It  has  a  conical  shape  and  is  called 

a  cone  or  pencil  of  lights  regardless  of  the  shape  of  the  opening 

through  which  it  passes.     When 

the  source  of  the  light  is  at  a 

relatively    great    distance,    any 

small   area  of  a  wave  front  is 

sensibly  plane  (Fig.  166),  and 

the  light  that  passes  through  a 

small  opening  is  cylindrical  in 

shape  and  is  called  a  beam  of 

light  (represented  by  CEFD  in 

the  figure) .    A  pencil  or  a  beam 

of  light  so  slender  that  its  cross-section  has  no  appreciable  area 

is  commonly  called  a  ray  of  light.     A  ray  is  often  regarded  merely 

as  a  mathematical  line  indicating  a  direction  in  which  light  travels. 

In  this  sense  any  radius  of  a  train  of  spherical  waves  is  a  ray.     Rays, 

in  either  sense  of  the  word,  are  perpendicular  to  the  wave  fronts. 

333.  Why  Light  travels  in  Straight  Lines. — The  thoughtful 
pupil  will  perhaps  wonder  why  light  does  not  spread  out  after 
going  through  an  opening  or  past  the  edge  of  an  object.  We 
know  that  sound  travels  round  buildings  and  other  obstacles.  If 
light  is  a  wave  motion,  why  does  it  not  do  the  same?  It  was  the 
want  of  an  answer  to  this  question  that  caused  Newton  and  his 
followers  to  reject  the  wave  theory. 

After  more  than  a  century,  experiments  were  devised  which 
proved  that  this  difference  between  the  behavior  of  light  and 
sound  is  due  to  the  very  great  difference  in  their  wave  lengths. 
The  waves  of  most  sounds  are  from  one  to  ten  feet  in  length ; 
light  waves  (which  are  also  capable  of  accurate  measurement)  vary 
from  33,000  to  64,000  to  the  inch.  It  was  found  that  light  waves 
and  sound  waves  behave  in  a  similar  manner  when  they  pass 
through  openings  or  encounter  obstacles  of  the  same  relative  size  in 
comparison  with  their  wave  lengths.  The  propagation  of  light  in 
straight  lines,  leaving  the  space  behind  opaque  objects  in  shadow, 
is  due   to  the  fact  that  ordinary  bodies  and  openings  are   enor- 


26o  Light 

mously  large  comj)ared  with  the  wave  length  of  light.  Sound 
l)ehaves  in  a  similar  manner  under  corresponding  conditions, 
forming  what  may  be  termed  sound  shadows.  "  Some  few  years 
ago  a  jx)wd^r  hulk  exploded  on  the  river  Mersey.  Just  opposite 
the  spot  there  is  an  opening  of  some  size  in  the  high  ground  which 
forms  the  watershed  between  the  Mersey  and  the  Dee.  The 
noise  of  the  explosion  was  heard  through  this  opening  for  many 
miles,  and  great  damage  was  done.  Places  quite  close  to  the 
hulk,  but  behind  the  low  hills  through  which  the  opening  passes, 
were  completely  protected,  the  noise  was  hardly  heard,  and  no 
damage  to  glass  and  such  like  happened.  The  opening  was  large 
compared  with  the  wave  length  of  the  sound  "  —  Glazebrook's  Phys- 
ical  Optics, 

When  light  passes  through  an  opening  that  is  not  large  com- 
pared wth  the  wave  length  (as  is  generally  the  case  with  sound), 
it  spreads  out  into  the  region  that  is  ordinarily  occupied  by  the 
shadow.  A  simple  example  of  these  effects  is  obtained  by  look- 
ing through  a  handkerchief,  held  close  to  the  face,  at  a  brightly 
illuminated  pinhole  or  narrow  slit  about  the  width  of  a  pin.  (Try 
it.)  The  pinhole  or  slit  may  be  made  in  a  piece  of  cardboard, 
and  illuminated  by  holding  it  in  front  of  a  lamp  or  gas  jet.  The 
light  spreads  out  in  different  directions  in  passing  through  the 
narrow  spaces  between  the  threads  of  cloth,  causing  the  slit  to 
look  like  a  number  of  parallel  slits,  and  the  hole  like  a  square 
pattern  of  many  holes.  Phenomena  of  this  class  are  studied  in 
advanced  physics.  It  is  by  means  of  experiments  involving  such 
phenomena  that  the  wave  lengths  of  light  are  determined. 
Laboratory  Exercise  -//. 

334.  Shadows.  —  A  cone  of  light  is  intercepted  by  an  opaque 
body,  as  ab  (Fig.  167),  when  the  source 
of  the  light,  Z,  is  so  small  that  it  may  be 
regarded  as  a  point;  and  the  light  is 
wholly  excluded  from  the  portion  of  this 
6"'^^^^^B  conical  space  that  lies  beyond  the  body. 
Fig.  167.  d   When  the  source  of  light  is  of  appreciable 


Nature  and  Transmission  of  Light      261 


size,  the  light  is  wholly  excluded  from  a  portion  of  the  space 
beyond  the  body,  as  acdb  (Fig.  168)  ;  and  this  space  is  surrounded 
or  enveloped  by  a  space  from  which  the 
light  is  partially  excluded.  The  latter  space 
receives  light  from  a  part  of  the  source, 
the  light  from  the  remainder  being  inter- 
cepted by  the  object. 

In  physics  the  word  shadow  means  the 
space  from  which  light  is  wholly  or  partly 
excluded  by  an  opaque  body.  The  dark 
area  upon  any  surface  where  it  intercepts  a 
shadow  is  a  cross-section  of  the  shadow,  and  should  be  so  named. 
The  part  of  the  shadow  from  which  the  hght  is  wholly  excluded  is 
called  the  umbra;  the  partly  illuminated  space  surrounding  the 
umbra  is  called  the  penumbra.  The  penumbra  merges  impercep- 
tibly into  fully  illuminated  space  at  its  outer  surface  ;  the  boundary 
between  the  penumbra  and  umbra  is  more  sharply  defined. 

335.    Solar  Eclipses.  —  In   Fig.   169,  6"  represents  the  sun,  ^ 
the  earth,  M^  the  moon  at  new  moon,  and  M^  at  full  moon.     The 


Fig.  168. 


■'V 


Fig.  169. 

shadows  of  the  earth  and  the  moon  are  diminishing  cones,  termi- 
nating in  a  point.     (Why?) 

On  account  of  the  varying  distances  of  the  sun  and  the  moon 
from  the  earth,  the  moon's  shadow  {i.e.  the  umbra)  sometimes 
reaches  the  earth  and  is  sometimes  too  short  to  do  so.  The  cross- 
section  of  the  moon's  shadow  is  never  more  than  167  miles  wide 
at  the  earth's  surface.  Within  the  shadow  the  sun  is  totally 
eclipsed ;  within  the  penumbra,  which  covers  a  much  larger  area, 
the  eclipse  is  partial. 

It  is  evident  that  an  eclipse  of  the  sun  can  occur  only  at  new 
moon  ;  but  there  is  not  an  eclipse  at  every  new  moon,  for  the  moon 


262  Light 

generally  passes  to  one  side  or  the  other  (above  or  below  the  plane 
of  the  paper  in  the  figure)  of  the  straight  line  between  the  sun  and 
the  earth.  The  least  number  of  solar  eclipses  that  can  occur  in  a 
year  is  two,  and  the  greatest  number  is  five. 

336.  Lunar  Eclipses.  —  The  diameter  of  the  earth's  shadow  at 
the  distance  of  the  moon  is  about  two  and  two  thirds  times  the 
diameter  of  the  moon.  When  the  moon  passes  entirely  into 
the  shadow,  it  is  totally  eclipsed  ;  when  only  one  side  of  it  passes 
through  the  shadow,  the  eclipse  is  partial.  There  is  no  perceptible 
dimming  of  the  moon  within  the  penumbra  until  it  almost  reaches 
the  shadow.  An  eclipse  of  the  moon,  either  total  or  partial,  is  of 
course  visible  to  half  the  earth  simultaneously.  Since  the  moon  is 
a  nonluminous  body,  shining  only  by  reflected  sunlight,  it  would 
be  invisible  when  totally  eclipsed  if  it  were  not  for  the  fact  that 
some  light  is  bent  out  of  its  course  (refracted)  into  the  shadow 
in  passing  through  the  earth's  atmosphere.  The  moon  is  thus 
illuminated  with  a  dull,  copper-colored  light. 

An  eclipse  of  the  moon  can  occur  only  at  -ftrfHnobn ;  but  the 
moon  generally  escapes  the  shadow  by  passing  to  one  side  or  the 
other  of  it.  The  number  of  lunar  eclipses  in  a  year  varies  from 
none  to  three. 

337.  Images  produced  by  Small  Openings.  —  When  light  from 
any  luminous  or  brightly  illuminated  object  falls  upon  a  screen  after 
passing  through  a  minute  opening,  such  as  a  pinhole,  it  forms 
upon  the  screen  an  inverted   image  of  the  object.     Figure  170 

shows  how  such  images  are 
produced.  Every  point  on 
the  surface  of  the  object 
is  a  source  of  light  which 
travels  outward  in  all  direc- 
tions from  the  surface.  A 
sle  Jer  cone  of  this  light 
from  each  point  passes  through  the  pinhol  and  illuminates  a  small 
spot  on  the  screen  of  the  same  shape  as  the  opening.  Since  these 
spots  have  the  same  relative  positions  as  the  corresponding  points  of 


Nature  and  Transmission  of  Light      263 

liie  object,  and  are  illuminated  by  light  of  the  same  color  as  those 
points,  they  unite  into  an  image  which  reproduces  the  form  and 
color  of  the  object.  The  inversion  of  the  image  is  due  to  the 
crossing  of  the  cones  of  light  at  the  opening.  If  the  opening  is 
very  small,  the  image  is  quite  sharply  defined,  but  faint.  With  a 
larger  opening,  the  image  is  brighter,  but  poorly  defined  (blurred) ; 
for  the  light  from  each  point  of  the  object  now  covers  a  larger  spot 
pn  the  screen,  and  these  spots  overlap  more  and  more  as  their  size 
increases.^ 

338.  Intensity  of  Illumination.  —  The  intensity  of  illumination 
on  any  surface  is  the  quantity  of  light  received  per  unit  area  of  the 
surface. 

Effect  of  Distance, —  The  intensity  of  the  light  from  any  source, 
or  the  intensity  of  the  illumination  that  it  produces  on  any  surface, 
is  inversely  proportional  to  the  square  of  the  distance  from  the 
source.  This  "  law  of  inverse  squares  "  is  the  same  as  that  for 
sound  (Art.  288),  and  for  the  same  reasons.  The  reasoning,  as 
applied  to  light,  may  be  briefly  stated  as  follows:  (i)  Assuming 
that  there  is  no  loss  of  light  by  absorption  or  other  cause,  the 
same  total  quantity  of  light  (radiant  energy)  passes  all  cross- 
sections  of  a  cone  of  light ;  (2)  hence  the  quantity  of  light  per 
unit  area  at  any  distance  is  inversely  proportional  to  the  area  of 
the  cross-section  of  the  cone  at  that  distance  ;  (3)  but  the  area  of 
the  cross-section  of  a  cone  of  any  shape  is  proportional  to  the 
square  of  the  distance  from  its  vertex  (Fig.  171);  (4)  hence  the  law. 

1  If  the  class  room  can  be  made  perfectly  dark,  an  interesting  study  of  these //'«- 
hole  images,  as  they  are  called,  can  be  made  by  admitting  light  into  the  darkened 
room  through  a  small  hole  in  a  window  screen  or  shutter,  and  catching  it  up>on  a 
screen  of  oiled  tissue  paper  or  upon  the  opposite  wall  of  the  room.  In  the  latter 
case  the  opening  must  be  a  centimeter  or  more  in  diameter  to  give  suflficent  illumi- 
nation. Under  these  conditions  an  image  of  the  landscape  in  its  natural  colors  will 
be  formed  upon  the  wall  or  the  screen.  A  pinhole  camera  can  be  made  from  a 
pastelxjard  box  a  foot  or  more  in  length.  A  large  pinhole  is  made  in  the  center  of 
one  end  to  admit  the  light,  which  is  caught  upon  a  screen  of  oiled  tissue  paper 
pasted  in  position  across  the  center  of  the  box.  A  hole  half  a  centimeter  or  more 
in  diameter  is  made  in  the  other  end  of  the  box.  The  image  upon  the  screen  is 
viewed  by  placing  the  eye  close  to  this  hole. 


264  Light 

Effect  of  Illuminating  Power. — The  total  quantity  of  light  con- 
tinuously given  out  by  any  source  is  called 
its  illuminating  power.  The  light  given 
out  by  a  standard  candle  is  taken  as  the 
unit  and  is  called  a  candle  power.  Thus  an 
incandescent  lamp  of  sixteen  candle  power 
gives  out  sixteen  times  as  much  light  as 
a  standard  candle.  The  intensities  of  iHumination  produced  by 
two  sources  of  light  at  equal  distances  are  proportional  to  the 
illuminating  powers  of  the  sources. 

Effect  of  Distance  and  Illuminating  Poiver  Combined.  —  Let  P 
denote  the  illuminating  power  of  a  source  of  light,  and  /  the  in- 
tensity of  illumination  which  it  produces  at  the  distance  D;  then, 
since  /  is  proportional  to  P  and  inversely  proportional  to  the 
square  of  Z>,  we  may  write  p 

^^  D"' 

339.  Photometry.  —  Photometry  deals  with  the  comparison  and 
measurement  of  the  illuminating  powers  of  different  sources  of  light. 
Any  apparatus  by  means  of  which  such  measurements  are  made 
\%Z2^^^2i  photometer.  We  can  form  no  reliable  estimate  of  the 
relative  brightness  of  unequally  illuminated  surfaces,  but  are  able 
to  judge  with  considerable  accuracy  whether  two  adjacent  parts  of 
the  same  surface  are  equally  illuminated  ;  hence,  with  all  forms  of 
photometers,  the  distances  of  the  lights  compared  are  adjusted  to 
give  equal  illumination. 

Let  Px  and  P^  denote  the  illuminating  powers  of  two  sources  of 
light,  and  let  /,  and  L  denote  the  intensities  of  illumination  which 
they  produce  at  the  distances  D^  and  Z>2,  respectively ;  then, 

If  the  distances  are  such  that  the  intensities  of  illumination  are 
equal  {i.e.  if  I^  =  /a),  then 

'  —  ^^2     f.j.— p.p..r)'i.r)2 


Nature  and  Transmission  of  Light      265 

This  is  the  relation  used  in  photometric  measurements.  Stated 
in  words  :  The  illuminating  powers  of  two  sources  of  light  are  pro- 
portional to  the  squares  of  the  distances  at  which  they  produce 
equal  illumination. 

Laboratory  Exercise  48. 

340.  The  Shadow  Photometer  (Rumf ord's  Photometer) .  —  There 
are  several  forms  of  photometers ;  but  the  shadow  photometer, 
devised  by  Count  Rumford,  will  serve  as  a  sufficient  illustration. 
Figure  172  shows  the  adjustment  of  the  apparatus  for  a  comparison 


172. 


of  the  illuminating  powers  of  a  lamp  and  a  candle.  The  rod,  Ry 
casts  two  shadows ;  c  is  due  to  the  candle,  and  /  to  the  lamp. 
Each  source  of  light  illuminates  the  shadow  due  to  the  other ; 
hence  when  they  are  placed  so  that  the  shadows  are  equally  dark 
{i.e.  equally  illuminated),  they  give  equal  illumination  at  their 
respective  distances  from  the  screen.  The  room  must  be  dark- 
ened, or  other  sources  of  light  will  make  the  shadows  too  foint  for 
accurate  comparison. 

341.  The  Velocity  of  Light.  —  The  velocity  of  light  in  a  vacuum 
and  in  air  is  186,000  miles  per  second  in  round  numbers,  —  a 
velocity  sufficient  to  encircle  the  earth  seven  and  one  half  times 
in  one  second.  Notwithstanding  the  difficulties  involved  in  the 
measurement  of  such  rapid  motion,  this  velocity  has  been  re- 
peatedly determined  with  consistent  results  by  two  experimental 
methods  ;  it  has  also  been  computed  by  two  independent  methods 


266  Light 

from  astronomical  data.  We  shall  consider  only  the  astronomical 
method  by  which  the  first  determination  of  the  velocity  of  light 
was  made. 

Jupiter,  the  largest  of  the  planets,  revolves  about  the  sun  in  an 
orbit  whose  diameter  is  about  five  times  that  of  the  earth,  the 
time  occupied  in  one  revolution  being  nearly  twelve  years.  In  Fig. 
1 73,  S  represents  the  sun,  £*  and  £  two  positions  of  the  earth  in 

©\  its  orbit,  y*  and  J  two  positions  of 

\-v^    Jupiter.     Jupiter   is   accompanied 
\T^       by  a  number  of  satellites  (moons), 
s  ]i         which  revolve  round  it  as  the  moon 

I         does  round  the  earth.    The  nearest 
/  of  these  satellites  is  shown  in  the 

pftfc  _  figure.       It    passes    through    the 

_  /  shadow  of  Jupiter  in  each  revolu- 

Fic.  173.    /  •'   * 

/  tion,  and  is  thus  eclipsed  at  regular 

intervals  of  42  hr.  28  min.  and  36  sec.  An  observer  does  not  see 
these  eclipses  when  they  occur ;  for,  after  the  satellite  has  entered 
the  shadow,  it  continues  visible  unAV  the  light  that  last  Uft  it 
reaches  the  earth.  If  the  eclipses  were  observed  from  any  constant 
distance,  they  would  be  seen  at  regular  intervals ;  for  they  would 
all  be  seen  the  same  length  of  time  after  their  actual  occurrence. 
For  example,  if  the  earth  remained  at  its  least  distance  from 
Jupiter,  each  eclipse  would  be  seen  35  min.  after  it  occurred,  that 
being  the  time  required  for  light  to  travel  the  intervening  distance, 
EJ\  if  the  earth  remained  at  its  greatest  distance,  EJ,  each 
eclipse  would  be  seen  51  min.  and  40  sec.  after  it  occurred,  and 
the  observed  intervals  between  the  eclipses  would  be  the  same  as 
before.  But,  while  the  earth,  in  its  annual  revolution,  is  receding 
from  Jupiter,  the  observed  interval  between  successive  eclipses 
exceeds  the  true  interval  by  the  time  required  for  light  to  travel 
the'  added  distance  due  to  the  earth's  motion  during  the  interval. 
While  the  earth  is  advancing*  toward  Jupiter,  the  conditions  are 
reversed  and  the  observed  intervals  are  less  than  the  true  intervals. 
Hence  if  the  eclipses  are  computed  ahead  for  one  year,  begin- 


Nature  and  Transmission  of  Light      267 

ning  when  the  earth  is  at  E  and  assuming  a  constant  interval,  the 
echpses  as  observed  fall  constantly  more  and  more  behind  until 
the  earth  is  at  E\  when  they  are  16  min.  and  40  sec.  late.  Dur- 
ing the  second  half  of  the  year,  the  intervals  are  less  than  the 
average,  and  the  loss  is  gradually  made  up;  so  that  when  the 
earth  has  arrived  at  E  the  eclipses  are  again  on  time. 

These  irregularities  in  the  observed  intervals  were  discovered 
and  explained  by  Roemer.  a  Danish  astronomer,  in  1675.  He 
announced  that  the  whole  apparent  retardation  of  the  eclipses 
from  E  to  E'  was  the  time  required  for  light  to  travel  across  the 
earth's  orbit,  which  is  known  to  be  a  distance  of  186,000,000 
mi.^     (Compute  the  velocity  of  light  from  the  given  data.) 

PROBLEMS 

1.  {a)  What  is  the  necessary  condition  for  a  shadow  without  a  penumbra? 
(^)  for  an  umbra  of  finite  length?  (r)  for  an  umbra  of  indefinite  length? 
(</)  for  a  cylindrical  umbra?     Make  a  section  drawing,  illustrating  each  case. 

2.  WTiy  are  shadows  as  we  see  them  in  nature  not  i)crfectly  dark  ? 

3.  What  is  the  apparent  shape  of  the  moon  two  or  three  days  after  new 
moon?  at  the  first  quarter?  between  the  first  quarter  and  full  moon?  Ac- 
count for  these  apparent  shapes  of  the  moon. 

4.  State  and  account  for  the  change  in  the  size,  brilliance,  and  sharpness 
of  outline  of  a  pinhole  image  (<i)  when  the  screen  upon  which  it  is  caught  is 
moved  farther  from  the  opening;  (/i)  when  the  size  of  the  opening  is  increased. 

5.  What  determines  the  ratio  of  the  size  of  a  pinhole  image  to  the  size 
of  the  object? 

6.  Two  sources  of  light  give  equal  illumination  at  loo  cm.  and  30  cm., 
respectively.     Hqw  do  they  compare  in  illuminating  power? 

7.  At  what  distance  will  a  i6-candle-power  lamp  give  the  same  intensity 
of  illumination  as  a  standard  candle  at  a  distance  of  70  cm.  ? 

1  Light  reaches  us  from  the  sun  in  8  min.  and  20  sec.  An  express  train  travel- 
ing at  the  rate  0(43  mi.  per  hour  would  require  254  years  for  the  journey.  Yet  in- 
conceivably great  as  this  distance  is,  it  is  insignificant  in  comparison  with  that  of 
the  fixed  stars,  the  nearest  of  which,  so  far  as  is  known,  is  so  far  away  that  light 
from  it  requires  3.5  years  to  reach  us.  Light  from  the  north  star  requires  about 
50  years  for  the  journey.  Indeed,  so  vast  are  stdlar  distances  that  astronomers 
regularly  use  the  light  year  (the  distance  that  light  travels  in  one  year)  as  the  unit 
in  which  to  express  them. 


268  Light 

8.  (tf)  The  cross-section  of  a  cone  of  light  is  S  scm.  at  a  distance  of  20  cm. 
from  its  vertex.  What  is  its  cross.section  at  a  distance  of  icx}  cm.  from  the 
vertex?  (6)  How  does  the  intensity  of  the  light  at  the  second  distance  com- 
pare with  that  at  the  first  ? 

II.    Reflection  of  Light 

342.  Regular  and  Irregular  Reflection.  —  The  reflection  of  light 
by  different  surfaces  may  be  studied  by  means  of  a  beam  of  sun- 
hght  admitted  into  a  darkened  room.  A  quantity  of  chalk  dust 
in  the  air  makes  the  path  of  the  light  plainly  visible.  A  plane 
mirror  reflects  the  beam  in  a  definite  direction,  which  depends 
upon  the  angle  at  which  the  mirror  is  held.  The  light  is  called 
incident  light  before  reflection,  and  reflected  light  after  reflection. 
The  angles  made  with  the  surface  of  the  mirror  by  the  incident 
and  reflected  beams  are  always  equal  (as  would  be  shown  by 
exact  measurement)  in  whatever  position  the  mirror  is  held. 
When  the  incident  beam  is  perpendicular  to  the  mirror,  the  re- 
flected beam  coincides  with  it  {Exp.). 

The  beam  reflected  by  a  mirror  forms  a  small  bright  spot  on 
the  wall ;  but,  when  the  light  is  reflected  by  a  piece  of  tin,  the 
illuminated  spot  is  larger,  fainter,  and  less  sharply  defined,  showing 
that  the  reflection  from  the  tin  is  less  regular.  When  the  beam 
falls  upon  a  sheet  of  writing  paper  or  a  piece  of  cardboard,  the 
reflected  light  is  said  to  be  irregularly  reflected  or  diffused  by  the 
paper.  It  will  be  observed  that  the  paper  is  brilliantly  illuminated 
by  the  sunbeam  and  can  be  distinctly  seen  from  all  parts  of  the 
room;  while-  the  surface  of  the  mirror  is  nearly" invisible,  even 
when  the  eye  is  in  the  path  of  the  reflected  beam   {Exp.). 

Although  a  sheet  of  writing  paper  may  appear  to  be  perfectly 
smooth,  a  hand  lens  is  sufficient  to  show  minute  irregularities 
covering  the  entire  surface.  These  irregularities  are  the  cause  of 
the  diffusion  of  light  by  the  paper  (Art.  345). 

343.  Definitions.  —  In  Fig.  174,  AB  represents  a  section 
through  a  mirror  whose  surface  is  perpendicular  to  the  plane  of  the 
paper;   C^  represents  a  slender  beam  (ray),  falling  upon  the 


Reflection  of  Light 


269 


mirror  at  iV  (called  \kiQ  point  of  incidence)^  and  ND,  the  reflected 
beam  ;   MN  is  the   perpendicular  or 
normal  to  the  reflecting  surfiice  at  the 
point  of  incidence. 

The  angle  between  the  incident  beam 
(or  ray)  and  the  perpendicular  to  the 
reflecting  surface  at  the  point  of  inci- 
dence is  called  the  angle  of  incidence  (angle  CNM,  or  / ) ;  and  the 
angle  between  the  reflected  beam  and  this  perpendicular  is  called 
the  angle  of  reflection  (angle  MND,  or  r). 

344.  Laws  of  Reflection.  —  When  light  is  reflected  from  polished 
surfaces,  the  angles  of  incidence  and  reflection  are  equal,  and  they 
are  in  the  same  plane.  Since  this  plane  contains  the  normal,  it 
is  evidently  perpendicular  to  the  reflecting  surface. 

These  laws  can  be  proved  by  direct  measurement,  and  can  also 
be  deduced  from  the  experimental  study  of  images  (Art.  347). 
The  reason  for  this  behavior  of  light  is  illustrated  in  part  by  Fig. 
175,  which  represents  the  reflection  of  a  beam  of  light,  CDFE, 

by  a  mirror,  AB.  Consider  the  wave 
that  would  be  represented  by  LMN 
if  it  did  not  meet  the  mirror.  The 
edge  N  met  the  mirror  first,  was 
reflected,  and  is  now  at  N\  having 
traveled  since  reflection  a  distance 
DN'  equal  to  DN.  The  parts  of 
the  wave  from  N  Ko  M  were  succes- 
sively reflected  in  like  manner,  and  now  occupy  the  position  N^M. 
The  laws  of  reflection  are  the  result  of  the  fact  that  each  portion 
of  the  wave  travels  as  far  after  reflection  as  it  would  have  traveled 
in  the  same  time  if  it  had  not  met  the  reflecting  surface.  Hence 
in  the  figure  triangles  DN'M  and  DNM  are  congruent ;  from 
which  it  is  easy  to  prove  that  the  angles  of  incidence  and  reflec- 
tion are  equal.     (The  proof  is  left  to  the  pupil.) 

345.  Diffusion  of  Light. — The  irregular  reflection  or  diffiision 
of  light,  as  already  stated,  is  due  to  microscopic  irregularities  of 


Fig.  175. 


270 


Light 


Fu;.  176. 


the  reflecting  surface.  This  is  illustrated  by  Fig.  1 76,  which  repre- 
sents a  highly  magnified  section  of  an  unpolished  surface.  Such 
a  surface  is  made  up  of  minute  parts  in- 
clined at  all  angles  to  one  another.  The 
parts  of  a  beam  of  light  meet  these  minute 
areas  at  different  angles,  and  are  reflected 
in  all  directions.  The  irregularities  on  an 
unjx)lished  surface  are  so  numerous  that 
any  spot  large  enough  to  be  seen  includes 
a  sufficient  number  of  them  to  reflect  the  light  in  all  directions ; 
so  that,  in  'effect,  rvery  point  of  the  surface  is  a  source  of  li^ht 
which  travels  outivard  in  all  directions  as  from  a  luminous 
body. 

The  light  received  from  the  sky  during  the  day  is  sunlight  that 
has  been  diffused  by  minute  particles  of  dust  in  the  air.^  'J'his 
diffusion  takes  place  principally  in  the  lower  regions  of  the  atmos- 
phere, where  the  dust  particles  are  the  largest  and  the  most 
numerous ;  hence  the  sky  is  much  darker  upon  high  mountains 
than  at  lower  altitudes.  Shadows  cast  by  objects  in  the  sunlight 
are  only  relatively  dark,  as  they  are  generally  quite  strongly 
illuminated  by  diffused  light  from  the  sky  and  from  surrounding 
objects. 

346.  The  Light  by  which  Objects  are  Seen.  —  Any  object,  either 
luminous  or  illuminated,  is  seen  by  means  of  the  cones  of  light 
that  enter  the  pupil  of  the  eye  from  all  points 
of  the  visible  portion  of  its  surface.  Figure 
177  represents  three  such  cones  of  light  from 
three  points  of  a  surface.  When  an  object  is 
viewed  directly  {i.e.  without  the  aid  of  mirrors 
or  lenses),  the  light  follows  a  straight  path  to 
the  eye,  and  each  point  of  the  object  is  seen  in  its  tnie  position 
at  the  vertex  of  the  cone  of  light  that  enters  the  eye  from  the 
point. 

1  The  process  is  illustrated  by  the  effect  of  chalk  dust  ia  the  path  of  a  beam  of 
light  in  a  darkened  room. 


Fk;.  177. 


Reflection  of  Light 


271 


Laboratory  Exercise  4g. 

347.  The  Image  of  a  Point  in  a  Plane  Mirror.  —  It  is  found  by 
experiment  that  the  image  of  a  point  in  a  plane  mirror  (from 
whatever  point  it  may  be  viewed)  is  on  the  perpendicular  to  the 
mirror  from  the  point,  and  is  as  far  behind  the  reflecting  surface 
as  the  point  is  in  front  of  it  (Lab.  Ex.  49). 

The  image  is  at  the  vertex  of  the  cone  of 
light  by  which  it  is  seen ;  it  is,  in  fact,  the 
apparent  source  of  this  light.  This  is  shown 
in  Fig.  178,  in  which  AB  is  the  mirror,  O 
the  point  object,  and  /  its  image.  The  image 
is  the  center  of  the  reflected  waves ;  hence 
the  light  after  reflection  travels  just  as  it 
would  if  the  image  were  its  real  source,  the 
mirror  being  removed. 

This  explains  why  light  regularly  reflected 
by  a  mirror  does  not  render  the  mirror  itself 
visible.  The  mirror  is  seen  by  the  very  small  amount  of  light 
that  is  diffused  by  its  surface.  A  good  mirror  is  therefore  nearly 
invisible  when  its  surface  is  perfectly  clean. 

Having  determined  by  experiment  the  relative  position  of  a 
point  and  its  image  in  a  plane  mirror,  the  laws  of  reflection  may 
be  established  as  follows :  The  experiment  proves  that  OI  is 
perpehdicular  to  AB  and  that  (9C=  CI.  Hence  triangles  NCO 
and  NCI  are  right  triangles  and  are  congruent ;  from  which  it  is 
easily  proved  that  the  angle  of  incidence  ONM  is  equal  to  the 
angle  of  reflection  MNE.  (The  completion  of  the  proof  is  left 
to  the  pupil,  also  the  proof  that  the  angles  lie  in  the  same  plane.) 

Conversely,  if  we  a'ssume  the  law  of  reflection  to  be  known,  the 
position  of  the  image  can  be  demonstrated  geometrically.  (An 
exercise  for  the  pupil.) 

348.  The  Image  of  an  Object.  —  Any  image  formed  by  a  plane 
mirror  is  so  situated  that  the  line  joining  any  point  of  the  object 
and  the  image  of  that  point  is  perpendicular  to  the  mirror  and 
is  bisected  by  it.     Hence,  to  locate  such  an  image  in  a  drawing, 


272 


Light 


perpendiculars  are  drawn  from  a  sufficient  number  of  points  of  the 
object  to  the  mirror  and  extended  equal  distances  behind  it.  The 
extremities  of  the  perpendiculars  are  the  corresponding  points  of 
the  image.  Thus  in  Fig.  179  the  ends  of  the 
image  C  and  D'  are  found  by  dropping  per- 
pendiculars from  C  and  D  according  to  the 
rule. 

To  construct  the  path  of  light  to  the  eye 
from  any  ix)int  of  the  image,  as  C,  draw  a  line 
from  that  point  to  the  eye,  and  from  the  point 
of  intersection  of  this  line  with  the  mirror 
draw  a  line  to  the  corresix)nding  point  of  the  object.  Behind 
the  mirror  the  lines  are  dotted  to  indicate  that  they  are  only 
apparent  paths  of  the  light.  Such  diagrams  are  commonly  simpli- 
fied by  drawing  only  a  single  line  to  represent  the  cone  of  light 
that  enters  the  eye  from  any  point. 

Images  formed  by  plane  mirrors  are  evidently  of  the  same  size 
as  the  objects.     They  are  erect  (unless  the  mirror  is  horizontal, 
as  the  surface  of  still  water)  ;  but  object  and  image  differ  as  the 
right  hand  differs  from  the  left. 
Laboratory  Exercise  ^o. 

349.   Images  by  Multiple  Reflection.  —  Images  formed  by  suc- 
cessive reflections  from  two  mirrors  are  called  multiple  images  or 
images  of  images.     The  formation  of  multiple  images  and  the  rule 
for   locating  them   in   diagrams  are 
illustrated  in  Fig.  180.     After  reflec- 
tion  from  the  mirror  AB,  the  path 
of  light  from  O  is  the   same   as   it 
would  be  if  the  image  /j  were  its  real 
source  (Art.  347).     Hence  the  por- 
tion of  this  reflected  light  that  falls 
upon  the  mirror  CD  forms  a  second 
image  h,  whose  position  is  the  same 
as  it  would  be  if  /j  were  the  object, 
the  mirror  AB  being  removed.     In  this  sense  I^  is  the  image  of 


U 


Fig.  180, 


Reflection  of  Light  273 

/i,  and  is  located  by  first  locating  /j,  then  treating  it  as  the  object 
whose  image  is  L, 

Light  after  reflection  from  AB  and  CD  apparently  comes  from 
A  If  some  of  this  light  again  falls  upon  AB,  a  third  image  /, 
will  be  formed,  whose  position  is  found  by  treating  I-i  as  the  object. 
With  the  angle  between  the  mirrors  as  shown  in  the  figure,  /,  is 
the  last  image  of  the  series ;  for,  since  it  lies  behind  the  plane  of 
the  mirror  CD,  none  of  the  light  coming  from  it  or  from  its 
direction  can  fall  upon  this  mirror.  The  smaller  the  angle  between 
the  mirrors,  the  greater  is  the  number  of  images  in  the  series ; 
when  the  mirrors  are  parallel,  the  series  is  indefinite,  being  limited 
only  by  the  gradually  failing  intensity  of  the  light.  There  is,  of 
course,  a  second  series  of  images,  formed  by  reflection  first  from 
CD,  then  from  AB,  etc.  This  series  is  not  represented  in  the 
figure. 

Since  the  light  by  which  each  image  is  formed  comes  from  the 
direction  of  the  preceding  image  of  the  series,  we  have  the  follow- 
ing rule  (illustrated  in  the  figure  for  /,)  for  constructing  the  path 
of  the  light  to  the  eye  for  any  image  :  Draw  a  line  from  the  image 
to  the  eye  {L,E  in  the  figure) ;  from  the  point  where  this  line 
intersects  the  mirror  draw  a  line  to  the  preceding  image  of  the 
series  (/i  in  this  case)  ;  from  the  point  where  this  line  intersects 
the  other  mirror  draw  a  line  to  the  next  preceding  image  or,  if 
this  is  the  end  of  the  series,  to  the  object.  The  path  of  the  light 
that  reaches  the  eye  is  thus  constructed  backward  without  the 
necessity  of  measuring  angles.  (Show  that  this  method  of  con- 
struction makes  the  angles  of  incidence  and  reflection  equal.) 

PROBLEMS 

1.  Sound  is  reflected  regularly  from  a  brick  or  a  stone  wall,  and  even  from 
cliffs  whose  faces  are  quite  irregular.  Show  by  a  comparison  of  wave  lengths 
that  these  facts  are  not  inconsistent  with  the  irregular  reflection  of  light  from 
unpolished  surfaces. 

2.  (rt)  In  what  respects  does  a  pinhole  image  differ  from  an  image  in  a 
plane  mirror  ?  {F)  What  purpose  is  served  by  the  screen  upon  which  a  pin- 
hole image  is  caught  ?  {c)  Can  an  image  formed  by  a  plane  mirror  be  caught 
upon  a  screen  ? 


274 


Light 


3.  Why  are  stars  invisible  by  day  ? 

4.  Copy  Fig.  180  ami  locate  the  other  series  of  images.  Construct  the 
path  of  light  to  the  eye  for  the  third  image  of  either  series. 

5.  Draw  a  figure  of  mirrors  at  an  angle  of  90",  locate  the  images  of  a 
point  object,  and  construct  the  path  of  light  to  the  eye  for  each  image. 

6.  Make  a  similar  diagram  with  the  mirrors  at  an  angle  of  6o'\  and  the 
object  at  uneciual  distances  from  the  mirrors.  Prove  that  the  object  and 
images  lie  on  the  circumference  of  a  circle. 

7.  Make  a  similar  diagram  with  the  mirrors  parallel ;  locate  the  images 
of  each  series;  and  construct  the  path  of  light  to  the  eye  for  the  first  three 
images  of  one  series.     Prove  that  the  images  lie  in  a  straight  line. 

350.  Spherical  Mirrors.  —  A  spherical  mirror  is  a  mirror  whose 
reflecting  surface  is  a  portion  (usually  a  very  small  portion)  of  a 
spherical  surface.  It  is  concave  or  convex  according  as  the  reflec- 
tion takes  place  from  the  inside  or  the  outside  of  the  spherical 
surface.  Concave  mirrors  are  not  always  spherical ;  but  in  speak- 
ing of  them  it  is  generally  assumed  that  the  curvature  is  spherical 
unless  it  is  otherwise  stated.  A  spherical  mirror  is  represented  in 
a  section  diagram  by  an  arc  of  a  circle,  as  MN  (Fig.  181).     The 


Fig.  181. 


center  0/ curvature  of  a  spherical  mirror  is  the  center  of  the  sphere 
of  which  the  mirror  forms  a  part  (C  in  the  figure).  The  radius 
of  the  sphere  is  called  the  radius  of  curvature  of  the  mirror.  The 
angle  MCN,  formed  by  joining  the  center  of  curvature  and  extremi- 
ties of  the  mirror,  is  its  angular  aperture^  or,  simply,  the  aper- 
ture. The  straight  line  ACy  which  passes  through  the  center  of 
the  reflecting  surface  and  the  center  of  curvature,  is  called  the 


Reflection  of  Light 


275 


\ 


principal  axis  of  the  mirror  ;  any  other  line  through  the  center  of 
curvature  is  a  secondary  axis. 

351.  Reflection  of  a  Beam  by  a  Concave  Mirror. — The  same 
laws  of  reflection  hold  for  curved  surfaces  as  for  plane ;  but  the 
effect  of  the  reflection  is  different.  ^  When  a  beam  of  sunlight  is 
reflected  by  a  concave  mirror  in  a  darkened  room,  the  reflected 
light  is  seen  to  have  the  form  of  a  double  cone  {MFN  and  EFG^ 
Fig.  181).  The  light  is  reflected  as  a  converging  cone;  beyond 
the  point  of  convergence  (the  focus)  it  continues  as  a  diverging 
cone  {Exp.). 

In  the  study  of  reflection  from  plane  surfaces  we  met  with  only 
diverging  cones  of  light.  The  waves  of  a  diverging  cone  are 
convex  (viewed  from  the  side  toward  which  they  are  traveling) ; 
the  waves  of  a  converging  cone  are  concave  (see  figure).  If  the 
direction  of  propagation  of  a  diverging  cone  were  reversed,  the 
light  would  return  to  its  source  as  a  converging  cone  with  the  con- 
cave side  of  the  waves  in  advance.  '  A  converging  cone  of  light 
increases  in  intensity  toward  the  focus  for  the  same  reasons  that 
a  diverging  cone  increases  in  intensity  toward  its  source.  When 
the  sunlight  reflected  by  a  concave  mirror  is  caught  upon  white 
paper  at  the  focus,  it  is  intensely  bright,  and  the  temperature  may 
be  high  enough  to  set  fire  to  the  paper.  A  match  at  this  point  is 
quickly  ignited  (Exp.). 
^  The  convergence  of  the  light  reflected  from  a  concave  surfiice 
is  explained  by  the  laws  of  reflection.  Consider  the  light  incident 
at  il/when  the  incident  beam 
is  parallel  to  the  principal 
axis.  The  perpendicular  to 
the  mirror  at  M  is  MC,  a 
radius  of  the  spherical  sur- 
face. Since  the  angles  of 
incidence  and  reflection  are 
equal,  the  reflected  light  evi- 
dently crosses  the  princi- 
pal axis  at  some   point,  Fj  Fig.  182. 


276  Light 

between  the  mirror  and  the  center  of  curvature.  An  accurately 
constnicted  figure  will  show  that  light  incident  at  any  other  point, 
as  at  Hf  also  crosses  the  principal  axis  at  or  very  near  F  ;  hence 
all  the  reflected  light  converges  approximately  to  F, 

It  will  be  found  by  trial  that  the  position  of  the  focus  changes 
when  the  mirror  is  inclined  at  different  angles  to  the  beam.  When 
the  incident  beam  is  parallel  to  the  principal  axis,  the  focus  lies 
on  this  axis  ;  when  the  incident  beam  is  inclined  to  the  principal 
axis,  the  beam  and  the  focus,  F\  are  on  opposite  sides  of  it 
(Fig.  182). 
352.  The  Principal  Focus  and  the  Focal  Length.  —  Let  BAT 

(Fig.  183)  be  any  ray  parallel  to 
the  principal  axis  of  a  concave 
mirror,  and  M  the  point  of  in- 
cidence. C  is  the  center  of 
curvature,  and  MC  the  perpen- 
dicular at  the  point  of  incidence. 
F  is  the  point  where  the  reflected  ray  cuts  the  principal  axis. 

Angles  /  and  e  are  equal  (alternate-interior  angles  of  parallel 
lines),  and  angles  /  and  r  are  equal  (law  of  reflection).  Hence 
angles  r  and  e  are  equal,  the  triangle  MFC  is  isosceles,  and 
MF=FC.  If  MA  is  not  more  than  a  tenth  of  the  radius  of 
curvature  (which  must  be  the  case  if  the  angular  aperture  of  the 
mirror  is  not  above  8°  or  10°),  AF  is  very  nearly  equal  to  MF^ 
and  hence  also  to  FC  \  i.e.  AF=FC,  approximately.  The  aper- 
tures of  concave  mirrors  are  always  small  in  order  that  this  relation 
may  hold  for  light  incident  at  any  point  of  their  surface.  Hence 
all  rays  parallel  to  the  principal  axis  of  a  concave  mirror  are 
reflected  approximately  to  a  point  on  the  principal  axis  midway 
between  the  mirror  and  its  center  of  curvature.  This  point  is 
called  \.\\t  principal  focus  (F,  Fig.  183),  and  is  defined  as  the  point 
to  which  a  beam  of  light  parallel  to  the  principal  axis  converges 
after  reflection.  The  distance  from  the  mirror  to  the  principal 
focus  is  called  the  focal  length  of  the  mirror ;  it  is  one  half  the 
radius  of  curvature. 


Fig.  183. 


Reflection  of  Light  277 

Laboratory  Exercise  §1. 

353.  Conjugate  Foci.  —  Figure  184  shows  the  effect  of  a  concave 
mirror  on  light  radiated  from  a  luminous  point  O  on  its  principal 


Fig.  184. 

axis  beyond  the  center  of  curvature.  Since  the  angle  of  incidence 
of  the  ray  OM  is  less  than  that  of  a  ray  parallel  to  the  principal 
axis  and  incident  at  the  same  point,  the  reflected  ray  will  evidently 
cross  the  principal  axis  at  some  point  /  between  the  principal 
focus  and  the  center  of  curvature.  It  can  be  shown  that  (with  a 
mirror  of  small  aperture)  all  the  light  of  the  incident  cone  MON 
converges  after  reflection  very  approximately  to  the  same  point  /, 
which  is  therefore  the  focus  of  the  reflected  light  and  the  image  of 
the  source  O. 

This  image  can  be  seen  and  its  position  accurately  determined 
in  two  ways :  (i)  It  can  be  viewed  directly  from  any  point  within 
the  diverging  cone  of  light,  E/Gf  and  will  then  be  seen  in  its 
true  position,  if  the  eyes  are  directed  toward  it  (not  toward  the 
more  distant  mirror).  (2)  It  can  be  caught  upon  a  screen,  which 
should  be  held  so  as  to  intercept  as  little  as  possible  of  the  inci- 
dent light  MON.  When  the  screen  is  placed  at  /,  the  image 
appears  upon  it  as  a  bright  point ;  when  it  is  placed  at  either  a 
greater  or  a  less  distance  from  the  mirror,  it  is  illuminated  over  an 
area  which  is  the  cross-section  of  the  cone  of  reflected  light  2X 
that  distance  (see  the  laboratory  exercise). 

If  the  source  of  light  is  removed  from  O  to  /,  the  former  angles 
of  reflection  at  each  point,  as  at  M,  become  the  angles  of  incidence, 
and  vice  versa;  and  the  reflected  light  will  converge  to  O. 
Hence  the  positions  of  image  and  object  are  interchangeable. 
Any  two  points  which,  like   O  and  /,  are  so  situated  that  light 


2/8  Light 

radiating  from  either  converges  after  reflection  to  the  other,  are 
called  conjugate  foci.  Conjugate  foci  afford  an  ilhistration  of  the 
fact  that  light  can  always  traverse  the  same  path  in  both  directions. 

354.  Relative  Positions  of  Conjugate  Foci.  —  The  position  of 
the  focus  conjugate  to  any  given  focus  may  be  found  by  experi- 
ment or  determined  from  the  laws  of  reflection.  There  are  several 
cases,  as  follows  :  — 

{a)  Conjugate  Foci  on  the  Principal  Axis,  —  When  the  source 
of  light  is  a  luminous  point  on  the  principal  axis,  the  image  (con- 
jugate focus)  is  also  a  point  on  this  axis.     There  are  six  cases. 

(i)  When  the  object  is  indefinitely  far  away,  the  image  is  at  the 
principal  focus.  This  case  is  approximately  illustrated  by  a  sun- 
beam parallel  to  the  principal  axis ;  but  only  approximately,  since 
the  sun,  although  very  distant,  is  not  equivalent  to  a  point  source. 
The  light  at  the  focus  covers  a  small  round  spot  which  is  an  image 
of  the  sun. 

(2)  As  the  object  approaches  the  center  of  curvature  along 
the  principal  axis  from  a  great  distance,  the  image  moves  from 
the  principal  focus  toward  the  center  of  curvature  (Fig.  184). 

(3)  When  the  object  is  at  the  center  of  curvature,  the  image 
coincides  with  it ;  for  each  incident  ray  is  perpendicular  to  the 
mirror  and  retraces  its  path  to  the  center. 

(4)  As  the  object  moves  from  the  center  of  curvature  toward 
the  principal  focus,  the  image  recedes  from  the  center  of  curva- 
ture to  an  indefinite  distance.  This  is  the  equivalent  of  case  (2) 
with  the  object  and  image  interchanged. 

(5)  When  the  object  is  at  the  principal  focus,  the  image  is 
indefinitely  far  away,  or  we  may  say  that  no  image  is  formed; 
This  is  case  (i)  with  object  and  image  interchanged. 

(6)  As  the  object  is  brought  up  from  an  indefinite  distance 
to  the  principal  focus,  the  divergence  of  the  incident  cone  of  light 
(measured  by  the  angle  at  the  vertex  of  the  cone)  steadily  in- 
creases, and  the  convergence  of  the  reflected  cone  simultaneously 
decreases.  When  the  object  is  at  the  principal  focus,  the  reflected 
light  is  no  longer  convergent,  but  parallel. 


Reflection  of  Light 


279 


As  the  object  approaches  the  mirror  from  the  principal  focus, 
the  divergence  of  the  incident  hght  increases  still  further,  and  is 
now  only  partly  overcome  by  the  reflection.  The  reflected  light 
is  therefore  divergent;  and 
its  apparent  source  (or  im- 
age) is  a  point,  /  ( Fig.  185),  ,^,'" 
situated  l^ehind  the  mirror  ^"^^'^^ 
on  the  principal  axis  (pro-  """ 
duced).       The     image     is 

located  by  producing  back-  *  '  ^' 

ward  any  two  lines  representing  rays  of  reflected  light.  Such  an 
image  (or  focus)  is  called  virtual^  while  those  in  front  of  the  mirror, 
to  which  light  converges  after  reflection,  are  called  real.  As  the 
object  is  moved  up  to  the  mirror  from  the  principal  focus,  its 
image  (the  conjugate  focus)  approaches  the  mirror,  but  is  always 
at  a  greater  distance  from  it  than  the  object  is.     (Why?) 

(^)  Conjugate  Fofi  on  Secondary  Axes. —  Let  O  (Fig.  186) 
be  a  luminous  point  not  on  the  principal  axis  AC.     The  ray  (9C, 

passing  through  the  cen- 
ter of  curvature,  falls 
upon  the  mirror  perpen- 
dicularly, and  is  reflected 
back  along  the  same 
path.  The  image  of  O  must  therefore  be  at  some  point  on  this 
secondary  axis.  The  ray  OMj  parallel  to  the  principal  axis, 
passes  through  the  principal  focus  F  after  reflection.  The  point 
of  intersection  of  the  reflected  rays  NC  and  MF  is  /,  which  is 
therefore  the  point  of  convergence  of  ail  the  reflected  light  from 
O ;  i.e.  the  diverging  incident  cone  of  light  MON  is  changed  by 
reflection  into  the  converging  cone  MIN.  Conversely,  if  the 
object  were  removed  to  /,  J//A^  would  be  the  incident  and  MON 
the  reflected  light,  and  the  image  would  be  at  O. 

In  general :  When  the  source  of  light  is  a  luminous  point 
lying  on  one  side  of  the  principal  axis,  its  image  (the  conjugate 
focus)  lies  on  the  opposite  side  of  this  axis,  and  is  on  the  same 


28o  Light 

secondary  axis  as  the  object.      The  relative  positions  of  object 

and  image  on  a  secondary 

axis  are  the  same  as  when 

they   are   situated   on   the 

principal  axis.     Figure  187 

illustrates   the  case   where 

,<.l-:r-:''^^"!r^   I!  '      the   image   is    virtual,   the 

7— **  Fig.  187.  ,.  r  ,       ,  .       !    . 

distance  of  the  object  bemg 

less  than  the  focal  length  of  the  mirror. 

355.  Images  formed  by  Concave  Mirrors.  —  When  the  source 
of  light  that  falls  upon  a  concave  mirror  is  an  object  of  appreci- 
able size,  the  diverging  cone  of  light  from  each  point  of  the  object 
converges  after  reflection  to  the  corresponding  point  of  the  image  if 
the  image  is  real,  or  diverges  as  if  it  came  from  the  correspond- 
ing point  of  the  image,  if  the  image  is  virtual.  A  virtual  image 
formed  by  a  concave  mirror  is  in  this  respect  like  the  virtual 
images  formed  by  plane  mirrors. 

In  drawing  figures  to  illustrate  the  formation  of  images  by  con- 
cave mirrors,  the  size  and  position  of  the  image  are  determined  by 
finding  the  conjugate  focus  of  a  point  at  each  end  of  the  object. 
Since  the  conjugate  focus  of  any  point  is  the  point  of  intersection 
(real  or  apparent)  of  all  rays  from  the  point  after  reflection,  it  can 
be  determined  in  a  diagram  by  constructing  any  two  of  these  rays. 
This,  in  general,  requires  the  measurement  of  angles  of  incidence 
and  reflection ;  but,  unless  the  point  is  on  the  principal  axis,  there 
are  three  rays  from  it  whose  directions  after  reflection  are  known, 
and,  by  choosing  any  two  of  them,  the  measurement  of  angles  is 
avoided.  These  rays  are  as  follows  :  (i)  The  ray  (from  the  chosen 
point  of  the  object)  through  (or  from  the  direction  of)  the  center 
of  curvature.  This  is  reflected  back  along  the  same  path. 
(2)  The  ray  parallel  to  the  principal  axis.  This  passes  through 
the  principal  focus  after  reflection.  (3)  The  ray  through  the 
principal  focus.     This  is  reflected  parallel  to  the  principal  axis. 

This  method  of  construction  is  employed  in  Figs.  188,  189,  and 
190  in  locating  the  ends  of  the  imige  of  an  arrow.    These  figures 


Reflection  of  Light 


281 


.  illustrate  and  explain  the  characteristics  of  the  image  for  difierent 
positions  of  the  object.  When  the  distance  of  the  object  from 
the  mirror  is  greater  than  the  focal  length,  the  image  is  real  and 
inverted^  as  in  the  first  four  cases  following. 

(i)  When  the  distance  of  the  object  is  so  great  that  the  rays 
from  any  point  of  it  are  sensibly  parallel  (a  distance  not  less  than 
one  hundred  times  the 


focal     length    of    the 

mirror),  the  image  is 

at  the  principal  focus 

and   is  very   small  in 

comparison    with    the 

object.     In  Fig.  188  the  rays  A  and  A^  are  from  the  same  point 

at  the  top  of  the  object,  and  B  and  B^  from  the  same  point  at 

the  bottom  :  ab  is  the  image.     (As  the  object  is  at  a  great  distance, 

it  cannot  be- shown  in  the  figure.) 
/    (2)    When  the  object  is  beyond  the  center  of  curvature  but  not 

at  a  great  distance,  the 
image  is  between  the  center 
of  curvature  and  the  princi- 
pal focus,  and  is  smaller 
than  the  object  (Fig.  189). 
(3)    When  the  object  is 

at  the  center  of  curvature,  the  image  is  also  at  the  center  of 

curvature,  and  is  of  the  same  size  as  the  object.     (Draw  figure.) 

(4)  When  the  object  is  between  the  center  of  curvature  and 
the  principal  focus,  the  image  is 
beyond  the  center  of  curvature, 
and  is  larger  than  the  object 
(Fig.  189,  taking  ab  as  the  ob- 
ject, and  AB  as  the  image). 

(5)  When  the  object  is  at  the 
principal   focus,   the   rays   from 
any  point  of  it  are  parallel  after  reflection,  and  there  is  no  per- 
fect image  (Fig.  188,  taking  ab  as  the  object). 


Fig.  189 


282 


Light 


Fig.  19X. 


(6)  When  the  object  is  between  the  principal  focus  and  the 
mirror,  the  image  is  virtita/  and  /•/rr/,  and  is  larger  than  the  object 
(Fig.  190).  As  the  object  approaches  the  mirror,  the  image  also 
approaches  it  and  grows  smaller. 

356.  Spherical  Aberration  :  Parabolic  Mirrors. —  If  the  angular 
aperture  of  a  spherical  mirror  is  large,  the  light  that  falls  upon 

different  portions 
of  its  surface  from 
the  same  point  of 
the  object  is  not 
nil  brought  to  the 
same  focus  after 
reflection  (Fig. 
191).  T\\\s  aber- 
ration or  devia- 
tion r)f  the  light 
from  the  true  focus  increases  as  the  distance  of  the  point  of  inci- 
dence from  the  axis  increases ;  hence  the  images  formed  by  mir- 
rors of  large  aperture  are  very  imperfect.  If  the  aperture  (angle 
MCN)  does  not  exceed  10°,  the  focusing  is  approximately  perfect. 

When  reflectors  of  large  aperture  are  re- 
quired, as  in  searchlights  and  the  headlights  of 
locomotives,  parabolic  mirrors  are  used  (Fig. 
192).  The  curvature  of  a  parabolic  mirror 
gradually  decreases  toward  the  edge,  and  light 
radiating  from  the  principal  focus  is  reflected  parallel  to  the  prin- 
cipal axis  from  all  parts  of  the  surface.     The  mirror  of  a  reflecting 

telescope  is  parabolic. 

357.  Convex  Mirrors. 
—  A  beam  of  light  paral- 
lel to  the  principal  axis 
of  a  convex  mirror  is 
reflected  as  a  diverging 
cone  whose  vertex  is  the 
point  on  the  principal  axis  midway  between  the  mirror  and  the 


F.G.  192. 


Fig.  193. 


Refraction  of  Light  283 

center  of  curvature  (Fig.  193).     This   point  is  called  the  prin- 
cipal focus  {Exp.), 

The  divergence  of  a  cone  of  light  is  always  increased  by  reflec- 
tion from  a  convex  mir- 
ror ;  hence  the  reflected 
light  is  always  divergent 
and  only  virtual  images 
are    formed    (Fig.    194). 


As  an  object  approaches  *  ^^^ 

a  convex  mirror  from  a  great  distance,  its  image,  which  at  first 
is  at  the  principal  focus  and  is  very  small,  approaches  the  mirror 
and  increases  in  size.  It  is  always  erect  and  smaller  than  the  object. 
(The  pupil  may  draw  figures  illustrating  the  above  statements.) 

PROBLEMS 

1.  (rt)  Account  for  the  difTerence  in  the  brilliance  ami  distinctness  of  a 
pinhole  image  and  a  real  inmge  formed  by  a  concave  mirror.  {h)  A  pinhole 
image  is  an  imperfect  real  image.     Why  real  ?     Why  imperfect  ? 

2.  A  pinhole  image  can  be  seen  only  when  caught  upon  a  screen.  Why 
can  it  not  be  seen  in  the  air  like  a  real  image  formed  by  a  concave  mirror  ? 

3.  Prove  that  the  divergence  of  a  cone  of  light  is  not  changed  by  reflection 
from  a  plane  mirror. 

4.  Why  do  plane  and  convex  mirrors  form  only  virtual  images  ? 

5.  What  are  the  essential  characteristics  of  a  virtual  image  ?  of  a  real 
image  ?  (The  answers  involve  a  consideration  of  the  convergence  or  diver- 
gence of  the  reflected  light. ) 

6.  Prove  that  the  lengths  of  object  and  image  (whether  real  or  virtual) 
are  proportional  to  their  distances  from  the  center  of  curvature  of  a  concave 
mirror. 

III.   Refraction  of  Light 

358.  Refraction.  —  When  a  ray  of  light  passes  obliquely  from 
one  transparent  medium  into  another,  it  is  bent,  or  refracted,  at 
the  surface  of  separation  of  the  two  media.  The  refraction  of  a 
beam  of  light  on  entering  water  from  air  can  be  shown  in  a  dark- 
ened room,  by  causing  the  beam  to  fall  upon  the  surface  of  water 


284 


Light 


contained  in  a  rectangular  glass  vessel  (Fig.  195).  The  path  of 
the  beam  in  the  water  is  more  distinct  if  the  water  is  clouded  with 
a  small-  quantity  of  an  alcoholic  solution 
of  mastic  or  a  little  milk.  It  will  be  ob- 
served that  the  refracted  beam  is  less 
oblique  to  the  surface  than  the  incident 
beam  ;  on  passing  from  air  into  waicr^ 
Fig.  195.  f^^j^f  j^  refracted  to^vard  the  perpendicular 

to  the  surface  at  the  point  of  incidence.  The  change  of  direction, 
or  deviation^  of  the  beam  on  entering  the  water  is  less  when  the 
beam  falls  less  obliquely  upon  the  surface ;  and  when  the  inci- 
dence is  perpendicular,  the  beam  enters  the  water  without  change 
of  direction  {Exp.). 

In  Fig.  196,  EOH  represents  a  ray  of  light  passing  from  air 
into  water  at  O.  EO  is  the  incident  and  OH  the  refracted  ray. 
MN  is  the  perpendicular  to  the  surface  at 
the  point  of  incidence.  The  angle  /  is 
called  the  angle  of  incidence ^  r  the  ajigle  of 
refraction^  and  d  the  angle  of  dexnation. 
(Give  a  definition  of  each  without  using 
letters  to  designate  the  lines  and  angles.) 

A  ray  of  light  passing  from  one  medium 
into  another  will  retrace  its  course  if  started 
back  along  the  same  path  (see  Art.  353,  end). 
Thus  if  HO  (with  the  arrowhead  reversed)  be  taken  to  represent 
an  incident  ray  passing  from  water  into  air,  OE  will  be  the 
refracted  ray,  and  the  angles  of  incidence  and  refraction,  as  shown 
in  the  figure,  will  be  interchanged.  On  passing  from  water  into 
air  J  tight  is  refracted  away  from  the  perpendicular. 
La  bora  tot y  Exercise  J2,  Part  I. 

359.  Some  Effects  of  Refraction.  —  An  object  under  water 
always  appears  to  be  above  its  true  position  when  viewed  through 
the  surface.  This  is  explained  by  Fig.  197.  The  cone  of  light 
that  enters  the  eye  from  any  point  of  the  object,  as  A,  is  refracted 
from  the  perpendicular  on  entering  the  air ;  and  its  apparent  source 


Fig.  196. 


Refraction  of  Light 


285 


is  the  vertex  of  the  refracted  cone  A',  —  a  point  vertically  above 

A.     All  the  light  that  enters  the  eye  from 

the  object  is  similarly  refracted,  and  the 

object  appears  to  be  vertically  above  its 

true  position.     Strictly  speaking,  what  is 

seen   is  not  the  object,  but   its   image 

formed   by   refraction.      The    apparent 

elevation  of  the  object  increases  as  it  is 

viewed  more  and  more  obliquely,  since 

the  deviation  increases  with  the  angle  of 

incidence  (Fig.  198). 

The  broken  appearance  of  a  straight  object,  as  a  pencil,  when 
partly  immersed  in  water  in  an  oblique  position,  is  similarly 
explained  (Fig.  199).  There  is  also  an  apparent  shortening  of 
the  immersed  part  of  the  object.     When  the  object  is  vertical,  it 


Vu,.  197. 


Fig.  198. 


Fig.  199. 


appears  to  be  shortened,  but  not  bent.  This  is  the  simplest  proof 
that  the  direction  of  the  apparent  displacement  is  not  obliquely 
upward  from  the  observer,  as  inaccurate  observation  often  seems 
to  indicate,  but  vertical. 

The  apparent  irregular  motion  and  changing  distortion  of  peb- 
bles and  other  objects  in  the  bed  of  a  brook  or  along  the  shore  of 
a  lake  are  due  to  the  constantly  varying  angle  of  incidence  at 
which  the  light  meets  the  surface  of  the  water,  as  ripples  and 
larger  waves  pass  over  it.  The  varying  angle  of  incidence  causes 
a  varying  deviation,  and  this  determines  the  apparent  position  of 
the  point  from  which  the  light  comes.  The  distortion  is  caused 
by  the  unequal  refraction  of  the  light  from  different  parts  of  the 


286  Light 


object.  A  similar  distortion  is  observed  when  objects  are  viewed 
through  common  window  glass,  the  surfaces  of  which,  are  more  or 
less  wavy.  Moving  the  head  from  side  to  side  causes  the  distor- 
tion to  change.     (Why?) 

360.  The  Sine  of  an  Angle.  —  In  a  right  triangle  the  ratio  of  a 
leg  to  the  hypothenuse  is  called  the  sine  of  the  angle  opposite  to 
that  leg.  Thus,  in  the  right  angle  ABC  (Fig.  200),  BC  :  BA  \s 
the  sine  of  the  angle  A,  and  CA  :  BA  is  the  sine  of  angle  B.  The 
usual  form  of  expression  is  — 

sine^  =  ^,  sine^  =  ^. 
BA  BA 

The  three  right  triangles  ABC,  AB'C,  and  AB"C'  are  similar ; 

hence  BC:BA=B'C  :  BA  =  B"C"  :  i9"^=sine  A.  From  which 
it  will  be  seen  that  the  sine  of  an  angle 
is  independent  of  the  size  of  the  triangle. 
The  sine  increases  as  the  angle  increases 
(up  to  90°),  /////  no/  proportionally. 
Angles  and  their  sines  are  very  nearly 
proportional  for  small  angles  ;  but,  as  an 
angle  approaches  90°,  it  increases  much 

more  rapidly  than  its  sine.     The  sine  of  0°  is  o,  the  sine  of  30°  is 

.5,  the  sine  of  45°  is  — r  or  .707,  and  the  sine  of  90°  is  i.     The 

V2 
sine  of  any  angle  can  be  found  from  a  table  of  sines. 

361.  The  Laws  of  Refraction. — The  following  laws  of  refrac- 
tion have  been  established  by  experiment  :  — 

I.  Whatever  the  angle  of  incidence^  the  ratio  of  the  sine  of  the 
angle  of  incidence  to  the  sine  of  the  angle  of  refraction  is  constant 
for  tlu  same  tivo  media,  hut  imries  with  different  media. 

II.  The  angles  of  incidence  and  refraction  lie  in  the  same  plane, 
and  are  on  opposite  sides  of  the  perpendicular  to  the  surface  at  the 
point  of  incidence. 

To  illustrate:  I^t  AOC  (Fig.  201)  represent  a  ray  of  light 
passing  from  air  into  water  at  O,  and  BD  the  perpendicular  to 


FlO.   20I. 


Refraction  of  Light  287 

the  surface  at  this  point.     AB  and  CD  are  drawn  perpendicular 
to  BD ;  hence  triangles  ABO  and  CDO  are 
right  triangles,  and 

•       .     AB        .    .  CD 

sine  /  =     —   and  sine  r  = • 

AO  CO 

Then,  according  to  the  first  law,  the  ratio 

AB     CD    .  ,    , , 

sine  t ;  sine  r,  or : ,  is  constant  for 

AO     CO 

all  angles  of  incidence ;  and,  according  to 

the   second   law,  angles  /  and  r  lie   in  the 

same   plane.      This   plane   is,   of  course,    perpendicular   to   the 

refracting  surface,  since  it  contains  the  perpendicular  BD. 

If  the  distances  AO  and  OCf  along  the  incident  and  refracted 

rays  respectively^  are  taken  equal  (as  they  are  in  the  figure),  the 

expression  for  the  ratio  of  the  sines  reduces  to 

sine  /■ :  sine  r=AB  :  CD. 

362.  Indices  of  Refraction.  —  The  ratio  of  the  sine  of  the  angle 
of  incidence  to  the  sine  of  the  angle  of  refraction  icdien  light  enters 
a  substance  from  a  vacuum  is  called  the  index  of  refraction  of  the 
substance.  The  methods  employed  in  elementary  physics  are  not 
accurate  enough  to  detect  any  difference  between  the  index  of 
refraction  from  a  vacuum  into  a  substance  and  from  air  into  the 
same  substance.     Hence  such  differences  may  be  disregarded. 

According  to  the  first  law  the  index  of  refraction  of  a  substance 
is  constant  for  all  angles  of  incidence.  It  differs  very  slightly  for 
light  of  different  colors. 

If  the  index  of  refraction  of  one  substance  is  greater  than  that 
of  another,  the  first  is  said  to  be  more  refractive  or  to  have  greater 
refractive  power  than  the  second.  It  will  be  seen  from  the  table 
(p.  288)  that  the  more  refractive  of  two  media  is  in  some  cases 
the  denser  and  in  others  the  rarer  of  the  two.  The  refractive 
power  of  a  substance  cannot  be  determined  from  its  density  or 
any  other  property ;  it  can  be  found  only  by  experiment. 


288 


Light 

Indices  of  Refraction 


Substance 

Index  of  Refraction 

Density  (g.  per  ccm.) 

Diamond 

Carbon  bisulphide 

Glass,  flint 

(ilass,  crown 

Glycerine 

Turpentine 

Chloroform 

Alcohol 

Ether   

Water 

2.47102.75 

1.68 

1.58  to  1. 61 

1.52  to  1.56 

1.47 

1.47 

1-45 

1.36 

136 

1.34 

1.00029 

1. 00 

3-5 

1.29 

3.00  to  3.32 

2.5 

1.27 

0.88 

1-52 

0.80 
0.72 
1. 00 

Air 

Vacuum 

0.00129 

When  light  passes  from  a  less  refractive  to  a  more  refractive 
medium  (as  from  water  into  glass),  it  is  bent  toward  the  perpen- 
dicular ;  when  it  passes  from  a  more  refractive  to  a  less  refractive 
medium,  it  is  bent  from  the  perpendicular.  The  less  the  difference 
between  the  indices  of  refraction  of  the  two  media,  the  less  will  be 
the  deviation  for  a  given  angle  of  incidence  ;  and  if  their  indices 
are  equal,  light  will  pass  from  one  to  the  other  at  any  angle  with- 
out deviation.  The  ratio  of  the  sine  of  the  angle  of  incidence  to 
the  sine  of  the  angle  of  refraction  when  Ifght  passes  from  any 
medium  into  another  is  called  the  relative  index  of  refraction  from 
the  former  medium  to  the  latter. 

Laboratory  Exercise  S3- 

363.  Cause  of  Refraction. — The  emission  theory  of  light  ex- 
plained refraction  upon  the  assumption  that  the  velocity  of  light  is 
greater  in  the  more  refractive  medium ;  according  to  the  wave 
theory  it  is  less.  The  velocities  of  light  in  air  and  in  water  were 
determined  experimentally  by  Foucault,  a  French  physicist,  in 
1850.  "He  found  the  velocity  in  water  to  he  less  than  in 
air;   from  that  moment  Newton's  emission  theory  was   dead." 


Refraction  of  Light 


289 


It  had,  indeed,  been  little  more  than  alive   for  a  quarter  of  a 
century. 

Refraction  may  be  explained  in  a  general  way  as  follows  :  Let 
AH  (Fig.  202)  represent  a  beam  of  light 
passing  from  air  into  some  more  refrac- 
tive substance,  as  glass.  When  the  inci- 
dent beam  is  oblique,  one  side  of  a  wave, 
as  B  of  the  wave  BD,  enters  the  second 
medium  before  the  other  and  is  retarded, 
while  the  opposite  side  continues  with 
undiminished  velocity  in  the  air.  Thus, 
while  one  side  of  a  wave  travels  from  B  to 
F,  the  other  side  travels  from  D  to  E,  and 
the  wave  swings  round  so  as  to  become  more  nearly  parallel  to  the 
refracting  surface. 

If  Vx  denotes  the  velocity  of  light  in  air  and  e'-j  its  velocity 
in  the  second  medium,  then  DE  :  BE=  7/,  :  fg.  Angle  DBE  is 
equal  to  the  angle  of  incidence,  and  angle  BEF  is  equal  to  the 
angle  of  refraction.     (Why  ?)     Hence,  since  triangles  DBE  and 

DF  Fi  f^ 

BEF  are  right  triangles,  sine/=   -—   and  siner  = ;    from 

BE  BE 


DE 


which  sine  / :  sine  r  =  - — -  = 


/' 


Hence  the  index  of  refraction 


BF     v^ 

of  a  substance  may  be  defined  as  the  ratio  of  the  velocity  of  light 
in  a  vacuum  (or,  very  approximately,  in 
air)  to  its  velocity  in  the  substance. 

364.  Construction  for  the  Refracted 
Ray.  —  Let  EO  (Fig.  203)  be  a  ray  of 
light  passing  from  air  into  water  at  O, 
The  construction  for  the  refracted  ray, 
taking  \  as  the  index  of  refraction  of 
water,  is  as  follows  :  Draw  MN  perpen- 
dicular to  the  surface  AB  at  the  point 

Fig.   203.  r    •  J  J     r 

of  mcidence ;  and  from  any  convenient 
point,   C,  on  the  incident  ray  draw  CD  perpendicular  to  MN. 


M 

D 

1  .  1  , 

y 

7 

0       ^ 

290  Light 

Liy  off  on  AB  a  distance  OF  equal  to  }  of  CD,  aod  construct 
FL  perpendicular  to  AB  and  parallel  to  MN.  With  (?  as  a 
center  and  a  radius  equal  to  (9C,  describe  an  arc  cutting  FL 
z\.  G.     OG  is  the  refracted  ray.     (Prove  it.) 

To  construct  the  refracted  ray  for  light  passing  from  air  into 
any  substance,  take  OF  of  such  length  that  CD  :  OF  is  equal  to 
the  index  of  refraction  of  the  substance.  To  construct  the  re- 
fracted ray  for  light  passing  from  any  substance  into  air,  take  OF 
of  such  length  that  OF  :  CD  is  equal  to  the  index  of  refraction  of 
the  substance. 

PROBLEMS 

For  the  following  constructions  take  -J  for  the  index  of  refraction  of 
crown  glass,  J  for  that  of  water,  and  ]}  (which  is  f  -r-  J)  for  the  relative  index 
of  refraction  from  water  into  cniwn  |{lass. 

1.  Construct  the  path  of  a  ray  of  light  from  air  into  water  for  an  angle  of 
incidence  (a)  less  than  30",  (*)  between  40°  and  50**,  (c)  between  80*^  and 

2.  Construct  the  path  of  a  ray  of  light  (r?)  from  crown  glass  into  air; 
(d)  from  air  into  crown  glass. 

3.  Construct  the  path  of  a  ray  of  light  («)  from  water  into  crown  glass; 
(*)  from  crown  glass  into  water. 

365.  Refraction  through  a  Plate  having  Parallel  Surfaces.  —  In 
Fig.  204,  FOO'F  represents  a  ray  of  light  passing  from  air  through 
a  glass  plate  into  air  again,  the  surfaces 
AB  and  CD  being  parallel.  The  angle 
of  incidence  at  O'  is  equal  to  the  angle 
of  refraction  at  O ;  hence  the  angle  of 
refraction  into  the  air  is  equal  to  the  first 
angle  of  incidence.  The  emergent  ray 
O'F  is  therefore  parallel  to  the  incident 
ray  £0,  but  the  two  are  not  in  the  same 
straight  line. 

iG.  204.  rj.^^   point    from   which    the    ray   comes 

will  appear  to  be  on  the  line  ^^;  hence  objects  viewed  obliquely 
through  a  glass  plate  are  apparently  displaced  to  one  side  of  their 


Refraction  of  Light  291 

true  position.  This  apparent  lateral  displacement  increases  with 
the  thickness  of  the  plate,  with  its  index  of  refraction,  and  with  the 
angle  of  incidence  (Lab.  Ex.  52). 

366.  Refraction  through  a  Prism.  —  When  the  surfaces  of  a 
medium  through  which  light  passes  are  planes  incHned  at  an  angle 
with  each  other,  the  medium  is  called  a  prism,  and  the  angle 
between  the  refracting  surfaces  is  called  the  refracting  angle  of  the 
prism. 

In  Fig.  205  the  triangle  represents  a  section  of  a  glass  prism. 
A  is  the  refracting  angle  ^ 

for  light  passing  through  /\ 

the  surfaces  ^^  and  y4  C     ^~>-^^  /     \  ^,j^ 

The  ray  EFGH  is  re-  """^-^^  /  \     ,-'-"'" 

fracted  toward  the  per-  ^^^Z--i7fr^'~-\ ^^" 

pendicular  MNoxi  enter-         _^^,-^^^/a^\        '^'^"^^x'^Ell 
ing  the  prism  and  from   '  ^^  ,  ^  ^^^ 

the  perpendicular  il/'iV'  '"  ^^* 

on  leaving  it.  The  deviations  d  and  (f  are  in  the  same  direction 
(away  from  the  refracting  angle)  ;  hence  the  total  deviation  is 
their  sum  and  is  measured  by  the  angle  KPH.  This  angle  is 
called  the  afig/e  of  ikviation. 

The  deviation  increases  with  the  refracting  angle  of  the  prism 
and  its  index  of  refraction ;  it  also  varies  with  the  angle  of  inci- 
dence, being  least  when  the  angle  of  incidence  is  such  that  the 
angle  of  emergence  is  equal  to  it.  The  deviation  varies  slightly 
for  light  of  different  colors,  producing  effects  which  are  considered 
later. 

The  apparent  source  of  the  ray  GH  is  some  point  on  the  line 
HL ;  hence  an  object  viewed  through  a  prism  is  apparently  dis- 
placed in  the  direction  of  the  refracting  edge  of  the  prism. 

Laboratory  Exercise  52,  Parts  II  and  III. 

367.  Partial  Reflection  and  Refraction.  —  In  general,  when 
light  meets  a  smooth  surface  separating  two  transparent  media, 
part  of  it  is  reflected  and  part  refracted.  A  number  of  illustrations 
are  familiar  where  air  is  one  of  the  media.     For  example,  the 


292 


Light 


greater  part  of  the  light  falling  upon  window  glass  passes  through 
it  by  refraction,  as  described  in  Art.  365  ;  but  a  considerable  por- 
tion of  it  is  regularly  reflected  from  the  front  surface,  forming  an 
image  of  its  source  as  in  a  plane  mirror.  Such  images  are  dis- 
tinctly visible  when  the  glass  is  backed  with  black  cloth  to  pre- 
vent the  transmission  of  light  from  the  opposite  side.  Similarly, 
light  falling  upon  the  surface  of  still  water  is  partly  reflected,  form- 
ing images  ;  but  the  greater  part  is  refracted  into  the  water. 

It  is  by  partial  reflection  and  refraction  that  a  number  of  images 
of  a  small,  bright  object,  as  a  candle  flame  or  a  lighted  match,  are 
formed  by  a  single  mirror  when  the  images  are  viewed  obliquely. 
(Try  it.)  Tiiese  multiple  images  are  especially  prominent  in 
mirrors  of  thick  glass.  Let  ABDC  (Fig. 
206)  represent  a  section  of  a  mirror  taken 
at  right  angles  to  the  reflecting  surface 
AH^  and  O  a  luminous  point  in  front  of 
the  mirror.  Part  of  the  incident  ray  OE 
is  reflected  from  the  front  surface  at  E^ 
forming  (or  helping  to  form)  the  image  /i. 
The  greater  part  of  the  incident  light 
enters  the  glass,  and  is  reflected  at  the  rear 
surface.  The  greater  part  of  this  light  is  refracted  into  the  air  at 
/%  forming  the  brightest  of  the  images  /, ;  the  remainder  is  inter- 
nally reflected  at  /%  is  again  reflected  at  the  rear  surface,  and 
the  greater  part  of  it  passes  into  the  air  at  G,  forming  the 
image  /j.  This  process  is  repeated,  forming  still  other  images ; 
but  their  number  is  limited,  as  the  light  rapidly  diminishes  in 
intensity  with  the  successive  reflections  and  refractions. 

368.  Total  Reflection.  —  When 
light  is  incident  in  the  more  re- 
fractive of  two  media  (as  when  it 
passes  from  water  into  air),  the 
angle  of  refraction  is  always  greater 
than  the  angle  of  incidence.  For  a 
certain   angle  of  incidence,  MOE  Fig.  207. 


/^ 

u 

/ 

A    "*' 

c/ 

^        /^             B 

"^£:=i^s^ssyi=-^=-£:s- 

'^KUtTrT^ 

M 

i=^"2^'  -  1 

Fig.  208. 


Refraction   of  Light  293 

(Fig.  207),  the  angle  of  refraction  is  90°,  and  the  refracted  ray 

is  parallel  to  the   surface  of  the  water. 

For  a  greater  angle  of  incidence,  as  angle 

MOF,  refraction  cannot  take  place,  and 

all  of   the   light   is    reflected    internally 

according  to  the  laws  of  reflection.     The 

ray  OFG  is  therefore  said  to  be  totally 

reflected  at  F.     Total  reflection  in  water   can  be   exhibited  by 

reflecting  a  beam  of  light  upward  through  water  in  a  rectangular 

glass  vessel  (Fig.  208)  {Exp.). 

The  angle  of  incidence  in  the  more  refractive  medium  for  which 
the  angle  of  refraction  is  90°  is  called  the  critical  angle.  When 
the  angle  of  incidence  is  less  than  the  critical  angle,  refraction  and 
partial  reflection  take  place ;  when  it  is  greater,  total  reflection 
occurs.  When  light  is  incident  in  the  less  refractive  medium, 
refraction  and  partial  reflection  take  place  at  all  angles  of 
incidence. 

The  critical  angle  for  water  and  air  is  48.5° ;  for  crown  glass 
and  air,  about  41° ;  for  flint  glass  and  air,  about  38°  ;  for  diamond, 
about  24°. 

Laboratory  Exercise  ^4. 

369.  Illustrations  and  Applications  of  Total  Reflection.  —  When 
a  glass  of  water  is  held  above  the  level  of  the  eye,  its  surface, 
viewed  from  below  through  the  side  of  the  glass,  looks  like  a 
mirror.  When  a  spoon  or  a  pencil  is  placed  in  the  glass,  the 
part  above  the  water  is  invisible,  but  a  very  distinct  image  of  the 
immersed  part  of  it  is  seen  in  the  surface.  This  image  is  formed 
by  total  reflection.     (Try  it.) 

Glass  prisms   afford   excellent   illustrations  of  total   reflection. 

Light  that  falls  upon  the  inside  of  any  face  of  the 

prism  at  an  angle  greater  than  the  critical  angle  is 

totally  reflected ;  and,  when  the  eye  is  in  position 

^^''"  to  receive  this  light  after  refraction  into  the  air,  the 

Fig.  209.        reflecting  surface  has  the  appearance  of  a  mirror 

and  forms  a  brilliant  image  of  the  source  of  the  light  (Fig.  209). 


294  Light- 

Prisms  having  an  angle  of  90°  and  two  angles  of  45°  are  used  in 
astronomical  telescopes  and  in  other  optical  instruments  to  change 
the  direction  of  the  light  by  90°  (Fig.  210).     Incident  light  per- 
pendicular to  the  face  AB  enters  the  prism  without 
deviation  and  meets  the  face  BC  2X  an  angle  of  45°. 
\^^|    /      Since  this  is  greater  than  the  critical  angle,  the  light  is 
\J*"-  totally  reflected  at  the  same  angle,  and  passes  out  of  the 
^     face  AC  without  deviation.     Such  prisms  are  the  most 
•10.  210.    pgj.fgj,^  mirrors  known.     They  give  only  a  single  reflec- 
tion, thus  avoiding  the  faint,  overlapping  images  due  to  multiple 
reflection  in  ordinary  mirrors.     A  totil-reflecting  prism  at  the  eye 
end  of  a  telescoi)e  adds   to   the   comfort  of  the  observer,  as  it 
enables  him  to  look  obliquely  tlownward  in  viewing  the  heavenly 
bodies,  instead  of  in  the  direction  in  which  the  telescope  points. 


IV.  Atmospheric  Refraction 

370.  Atmospheric  Refraction.  —  Although  the  refractive  power 
of  the  air  is  small,  it  gives  rise  to  a  number  of  interesting  and 
familiar  phenomena.  When  we  look  over  a  bonfire  or  a  hot  stove 
at  any  object  situated  beyond  it,  the  object  appears  to  undergo 
a  rapidly  changing  distortion,  similar  to  that  of  a  pebble  in  the 
bottom  of  a  brook.  This  appearance  is  due  to  the  colistantly 
changing  refraction  of  the  light  as  it  passes  through  the  currents 
of  air  rising  from  the  fire ;  for  these  currents  consist  of  bodies  of 
air  of  varying  temperatures  and  hence  of  varying  densities,  and  the 
refractive  power  of  air  increases  with  the  density.  A  similar  effect 
may  often  be  observed  when  the  line  of  sight  passes  near  the 
surface  of  some  object  that  has  become  hot  in  the  sunshine. 

371.,  Twinkling  of  the  Stars.  —  The  twinkling  of  the  stars  con- 
sists in  a  rapid,  irregular  variation  in  brightness.  With  the  aid  of 
a  telescope,  this  is  seen  to  be  accompanied  by  a  dancing  motion. 
The  phenomenon  is  wholly  atmospheric  ;  the  stars  themselves  are 
fixed  and  shine  with  a  steady  light.  The  dancing  is  a  rapid  change 
of  apparent  position,  caused  by  changing  refraction  as  currents  of 


Atmospheric   Refraction 


295 


air  of  varying  density  cross  the  line  of  sight.  As  a  beam  of  light 
passes  through  successive  layers  of  air,  the  refraction  at  the  irregu- 
lar boundaries  separating  thorn  may  cause  either  a  slight  con- 
vergence or  divergence  of  the  beam.  The  first  increases  the 
intensity  of  the  light,  the  second  diminishes  it ;  and  the  twinkling 
is  largely  due  to  the  rapid  alternation  of  these  effects.  Stars  near 
the  horizon,  the  light  from  which  traverses  a  greater  stretch  of 
atmosphere,  twinkle  more  than  those  overhead.  The  twinkling 
also  differs  greatly  on  different  nights  according  to  the  steadiness 
of  the  air. 

372.  Regular  Atmospheric  Refraction. — The  inconstant  or 
irregular  refraction  to  which  the  twinkling  of  the  stars  is  due  is 
small  in  comparison  with  the  regu- 
lar refraction,  due  to  the  increas- 
ing density  of  the  atmosphere 
from  its  upper  limit  to  the  earth's 
surface.  Light  traveling  obliquely 
downward  through  the  atmos- 
phere is  bent  continuously  toward 
the  perpendicular  (Fig.  211).  The  total  deviation  thus  produced 
varies  from  zero  for  heavenly  bodies  directly  overhead  to  a  little 
more  than  half  a  degrfte  at  the  horizon  (it  is  greatly  exaggerated 
in  the  figure). 

Since  the  angular  diameter  of  the  sun  at  the  earth  is  about  half 
a  degree,  the  sun  is  really  just  below  the  horizon  when  it  appears 
to  be  just  above  it.  Thus,  on  account  of  atmospheric  refraction, 
sunrise  occurs  from  two  to  four  minutes  earlier  than  it  otherwise 
would  (depending  upon  the  angle  that  the  sun's  path  makes  with 
the  horizon),  and  sunset  is  retarded  by  the  same  amount. 

373.  Mirage.  —  In  sandy  deserts  the  reflection  of  the  sky  and 
of  the  scattered  trees  and  other  objects  in  the  landscape  is  often 
seen  in  the  distance,  on  hot  sunny  days,  as  in  the  surface  of  a 
calm  lake.  This  optical  illusion  is  called  a  mirage.  It  is  due  to 
the  heating  and  expansion  of  the  air  in  contact  with  the  hot  sand  ; 
as  a  result  of  which  the  density  of  the  air  increases  upward  for 


Fig.  211. 


296  Light 

some  distance  from  the  ground.  Light  traveling  obliquely  down- 
ward through  this  layer  of  air  is  gradually  bent  from  the  perpen- 
dicular; and  the  angle  of  incidence,  if  nearly  90°  at  first,  may 
thus  become  greater  than  the  critical  angle.  The  light  is  then 
totally  reflected  by  the  layer  of  rarer  air  into  which  it  cannot  pass, 


and  is  refracted  toivard  the  perpendicular  as  it  returns  through 
the  denser  air  alx)ve.  By  this  total  reflection  images  are  formed 
like  those  seen  in  the  surface  of  still  water  (Fig.  212).  The  sky 
and  other  objects  are  also  seen,  at  the  same  time,  erect  and  in 
their  true  positions,  by  light  that  comes  straight  to  the  eye. 

V.  Lenses 

374.  Lenses.  —  A  lens  is  a  portion  of  a  transparent  medium 
bounded  by  two  curved  surfaces  or  by  a  plane  and  a  curved 
surface.  Lenses  are  usually  made  of  glass,  and  their  curved 
surfaces  are  usually  spherical.  There  are  six  forms  of  spherical 
lenses,  sections  of  which  are  represented  in  Fig.  213.  From 
their  optical  effects  they  are  classed  in  two  groups  of  three  each, 
as  follows :  — 

Convex  or  Convergins^  Lenses.  —  The  first  three  lenses  repre- 
sented in  the  figure  belong  to  this  class.     The  first  of  these  is 


Lenses 


297 


called  double  convex,  the  second  plano-convex^  and  the  third  con- 
cavo-convex. All  are  thicker  in  the  middle  than  at  the  edges. 
They  are  called  converging  lenses,  because  light  is  more  converg- 
ent or   less   divergent  after   passing  through   them  than   before 


Fio.  213. 

{Exp^.  The  three  forms  are  equivalent  in  their  effects  so  far  as 
the  purposes  of  elementary  physics  are  concerned  ;  and  the  double- 
convex  lens  having  surfaces  of  equal  curvature  is  the  only  one  that 
will  be  considered. 

Concave  or  Diverging  Lenses.  —  To  this  class  belong  the  last 
three  lenses  represented  in  the  figure.  The  first  of  these  is 
double-concave^  the  second  piano- concave ,  and  the  third  convexo- 
concave.  They  are  all  thinner  in  the  middle  than  at  the  edges ; 
and  are  called  diverging  lenses  because  they  increase  the  diverg- 
ence of  light  passing  through  them  {Exp.).  The  double-concave 
lens  will  be  taken  as  the  type  of  its  class. 

375.   The  Double-convex  Lens.  —  The  line  joining  the  centers 


of  curvature  of  the  spherical  surfaces  of  a  convex  lens  (C  and  C, 
Fig.  214)  is  called  iht  principal  axis  of  the  lens. 


298 


Light 


Let  OA  be  a  ray  of  light  from  a  luminous  point  on  the  prin- 
cipal axis.  On  entering  the  lens,  the  ray  is  bent  toward  the 
perpendicular  AC ;  on  ^merging  at  B,  it  is  bent  from  the  per- 
pendicular BC.  The  deviation  of  the  ray  is  the  same  as  it  would 
be  if  the  lens  were  replaced  by  a  prism  ADB  of  the  same  mate- 
rial and  having  faces  tangent  to  the  lens  at  A  and  B.  The  angle 
between  the  tangent  planes  at  the  jwints  of  entrance  and  emerg- 
ence of  a  ray  increases  toward  the  edge  of  the  lens  {g^.  angle  £ 
is  greater  than  angle  D)  ;  hence  the  deviation  also  increases.  A 
ray  of  light  traveling  along  the  principal  axis  falls  perpendicularly 
upon  both  surfaces  of  the  lens,  and  hence  passes  through  it  with- 
out deviation. 

The  increased  deviation  toward  the  edge  of  the  lens  is  almost 
exactly  what  is  required  to  bring  a//  the  refracted  light  to  the  same 
point,  /.     We  shall  for  the  present  regard  the  focusing  as  perfect. 

Thus  the  diverging 
cone  of  incident  light 
A'OG  undergoes  re- 
fraction at  the  surfaces 
of  the  lens  and  emerges 
as  a  converging  cone, 
L///,  forming  a  real 
'^'  ^'^*  image  of  its  source  at  /. 

The  effect  of  the  lens  is  also  shown  in  Fig.  215,  in  which  the 
curved  lines  represent  light  waves.  The  points  O  and  /  are 
called  conjitgaU  foci.  Light  radiating  from  either  converges  to  the 
other. 

376.  Conjugate  Foci  on  the  Principal  Axis.  — Real  Foci. — As 
the  luminous  point  6>(Fig.  214)  is  moved  from  the  lens  along  the 
principal  axis,  the  incident  cone  of  light  becomes  less  divergent 
and  the  refracted  cone  more  convergent ;  the  image  therefore 
moves  toward  the  lens  along  the  axis.  When  the  distance  of  the 
object  is  relatively  great  (not  less  than  one  hundred  times  the 
radius  of  curvature  of  the  surfaces  of  the  lens),  the  incident  light 
is  sensibly  parallel,  and  the  point  to  which  the  refracted  light  con- 


Lenses  299 

verges  is  called  the  principal  focus  of  the  lens  {F,  Fig.  216). 
There  is  another  principal  focus  at  the  same  distance  on  the  other 
side  of  the  lens,  correspond- 
ing to  an  incident  beam 
coming   from    the   opposite 

direction.     Since    light  can 

*u  *u    •  F'G.  216. 

traverse   the  same   path  m 

both  directions,  it  follows   that  light   radiating  from  a  luminous 

point  at  either  principal  focus  is  refracted  as  a  beam  parallel  to 

the  principal  axis. 

The  distance  of  the  principal  focus  from  the  lens  is  called  the 
focal  length  of  the  lens.  The  focal  length  depends  upon  the 
refractive  power  of  the  glass  of  which  the  lens  is  made,  as  well 
as  upon  the  curvature  of  its  faces.  It  can  be  shown  that,  when 
the  faces  have  equal  curvature  and  the  index  of  refraction  of  the 
glass  is  1.5,  the  focal  length  is  equal  to  the  radius  of  curvature. 
When  the  index  of  refraction  exceeds  1.5,  the  focal  length  is  less 
than  the  radius  of  curvature. 

As  the  luminous  point  is  moved  toward  the  principal  focus  from 
a  greater  distance,  the  incident  light  becomes  more  and  more  diver- 
gent and  the  refracted  light  less  convergent :  the  image,  therefore, 
recedes  along  the  axis.  When  the  object  is  at  the  principal  focus, 
the  light  is  refracted  as  a  parallel  beam,  as  stated  above,  and  the 
image  is  indefinitely  far  away. 

Virtual  Foci,  —  When  the  object  is  nearer  than  the  principal 
focus,  the  divergence  of  the  incident  light  is  greater  than  the 

lens  can  overcome,  and  the 
refracted  light  is  still  divergent, 


,^^^arx>--------- |^-p-|  «^ though  less  so  than  the  inci- 

^         '""'"'-^^^^^^^^^   dent   light   (Fig.    217).     The 

image  (or  focus)  is  therefore 
*  ^^^'  virtual  and   at  a  greater  dis- 

tance than  the  object.  As  the  object  is  moved  up  to  the  lens 
from  the  principal  focus,  the  image  approaches  the  lens  from  an 
indefinite  distance  on  the  same  side. 


300  Light 

377.  Conjugate  Foci  on  Secondary  Axes.  —  When  a  ray  of  light 
passes  obliquely  through  the  center  of  a  double-convex  lens,  the 
tangent  planes  at  the  points  of  entrance  and  emergence  of  the 

ray  (A  and  B,  Fig.  218) 
^  are  parallel.  The  emer- 
gent ray  lU  is  therefore 
parallel  to  the  incident 
rays  OA  (Art.  365). 
When  the  lens  is  thin  and  the  angle  of  incidence  small,  as  is  gen- 
erally the  case,  the  lateral  displacement  of  the  refracted  ray  is 
very  slight  and  may  be  disregarded.  Any  ray  through  the  center 
of  the  lens,  as  OABI,  is  therefore  regarded  as  a  straight  line,  and 
is  so  drawn  in  diagrams. 

Any  straight  line  through  the  center  of  a  lens  is  called  a  second- 
ary axis.  It  follows  from  the  above  that  the  conjugate  focus  of 
any  point  not  on  the  principal  axis  lies  on  the  secondary  axis 
through  that  point.  Conjugate  foci  on  secondary  axes  are  real  or 
virtual  under  the  same  conditions  as  for  foci  on  the  principal 
axis ;  i.e.  if  the  distance  of  a  point  is  greater  than  the  focal  length 
of  the  lens,  the  conjugate  focus  is  real ;  if  less,  it  is  virtual. 
Laboratory  Exercise  55. 

378.  Real  Images.  —  When  light  falls  upon  a  convex  lens  from 
an  object  situated  beyond  the  principal  focus,  the  diverging  cone 
of  light  from  each  point  of  the  object  converges  to  the  conjugate 
focus,  and  there  forms  the  corresponding  point  of  the  image, 
which  in  this  case  is  real.  The  image  can  be  caught  upon  a 
screen,  and  can  also  be  viewed  directly  from  any  point  in  the  path 
of  the  light  diverging  from  it.  Thus,  in  Fig.  214,  the  image  can 
be  seen  in  mid-air  from  any  point  within  the  cone  PIQ. 

There  are  three  rays  from  any  point  not  on  the  principal  axis 
whose  directions  after  passing  through  a  lens  are  known  :  (i)  The 
ray  through  the  center  of  the  lens  continues  in  the  same  straight 
line.  (2)  The  ray  parallel  to  the  principal  axis  is  refracted  so  as 
to  pass  through  the  principal  focus.  (3)  The  ray  through  the 
principal  focus  (on  the  same  side  as  the  object)  is  refracted  parallel 


Lenses 


301 


Fig.  219. 


to  the  principal  axis.  These  three  rays  are  shown  in  Fig.  218. 
In  drawing  figures,  the  conjugate  focus  of  any  point  can  be  deter- 
mined by  means  of  any  two  of  these  rays,  when  the  focal  length  is 
known,  without  constructing  angles  of  incidence  and  refraction. 
Figures  219,  220,  and  221  illustrate  this  method  of  construction. 
In  Fig.  219,  AB  may  be  regarded  as  the  object  and  ab  its 
image,  or  vice  versa.    Since 

any  point  of  the  object  and       a- ^ — ^  A-<^         b^ 

its  image  are  on  the  same 

straight   line   through  the 

center    of    the    lens,    the 

image,  if  real,  is  always  inverted.     Triangles  AOB  and  aOl>  are 

similar;    hence  the   lengths  (and   other  similar  dimensions)   of 

object  and  image  are  proportional  to   their  distances  from  the 

lens. 

The  following  consequences  of  these  geometrical  relations  are 
of  great  importance  in  optical  instruments :  — 

(i)  When  the  distance  of  the  object  is  so  great  that  the  rays 
from  any  point  of  it  are  sensibly  parallel,  the  real  and  inverted 

image  which  is  obtained  of 
it  is  at  the  principal  focus 
and  is  relatively  very  small 
(Fig.  220). 

(2)  When  the  object  is 
at  a  distance  only  slightly 
greater  than  the  focal  length,  its  image  is  relatively  distant  and 
greatly  enlarged  or  magnified.     (Draw  figure.) 

(3)  For  a  given  object  at  a  given  distance,  the  size  of  the  real 
image  increases  with  the  focal  length  of  the  lens ;  since,  under 
these  conditions,  the  greater  the  focal  length,  the  greater  is  the 
distance  of  the  image  from  the  lens  {Exp.).  (Illustrate  with  two 
drawings,  taking  lenses  of  unequal  focal  length.) 

(4)  When  the  object  is  at  a  relatively  great  distance  (Fig.  220), 
the  length  of  the  image  is  proportional  to  the  focal  length  of  the 
lens.     (Draw  figures  to  illustrate.) 


Fig.  220. 


-^^ 


302   v,o- 


X 


Light 


'--IM" 


379.  Virtual  Images.  —  When  the  distance  of  the  object  from 
the  lens  is  less  than  the  focal  length,  the  light  from  each  point  of 
the  object  is  still  divergent  after  refraction ;  and  its  apparent 
source  is  the  corresponding  point  of  the  image,  which  in  this  case 
is  virtual.  A  virtual  image  can  be  seen  only  by  looking  through 
the  lens  toward  the  object.  What  we  really  see  through  the 
lens  is  not  the  "  magnified  object,"  but  its  magnified  virtual 
image. 

Figure  221  illustrates  the  formation  of  a  virtual  image,  following 
„  the  usual  construction.     It  will 

be  seen  from  the  figure  that 

^"^irr^-^^.A        ^^      the    virtual    image    is    ahvays 

*p!^  erect  and  magnified,  and  is  at 

a  greater  distance  than  the  ob- 

::  ^.s---  ject.    A  similar  figure  in  which 

*"  ^^'  the  object  is  taken  nearer  the 

lens  will  show  that,  as  the  object  approaches  the  lens  from  the 

principal  focus,  the  image  also  approaches  the  lens  and  grows 

smaller.     (Draw  the  figure.) 

From  the  similar  triangles  A  OB  and  aOb  it  is  evident  that 
the  lengths  of  object  and  image  are  proportional  to  their  distances 
from  the  lens,  as  in  the  case  of  real  images.  The  less  the  focal 
length  of  the  lens,  the  larger  is  the  virtual  image  when  formed  at 
a  given  distance  from  the  lens.     (Draw  figure  to  illustrate.) 

A  convex  lens,  when  used  for  observing  the  enlarged  virtual 
images  of  minute  objects,  is  called  a  magnifying  glass  or  simple 
microscope. 

380.  Formulas  Relating  to  Convex  Lenses.  —  Formula  for 
Real  Images.  —  In  Fig.  222 

the   rays  AM  and  BN  are    '         "    ^- —        /IT""^^ ^^^^ 

drawn   as   if  they   were   re- 
fracted once  midway  between 
the  surfaces  of  the  lens  in- 
stead of  at  each  surface.     This  simplifies  the  present  problem, 
and  involves  no  appreciable  error. 


Lenses  303 

From  the  similar  triangles  A  OB  and  a  Ob, 

AB'.ahw  CO:  Oc. 
From  the  similar  triangles  MFN  and  aFb, 

MN'.ab-.'.OF'.Fc, 
Since  AB  =  MN^  we  have  from  these  proportions, 

CO'.Oc'.'.OF'.Fc, 
Let  CO  =  D  (the  distance  of  the  object),  Oc  =  d  (the  distance 
of  the  image),  and  OF=f  (the  focal  length  of  the  lens).     Sub- 
stituting these  values  in  the  last  proportion,  we  have 
D:d::f:{d-f). 
From  which  df^D{d-f), 

Transposing  and  combining, /(//  +  Z>)=  Dd, 

Dividing  by /?./5^,  .         ^^  =  7- 

JJa        J 

Separating  the  terms  of  the  fraction  and  reducing, 

-1  +  1  =  1. 
D^  d     f 
■f 
By  means  of  this  formula  any  one  of  the  three  quantities  Z>,  d^ 

and  /  can  be  found  when  the  other  two  are  given. 

Formula  for  Virtual  Images.  —  From  the  similar  triangles  fZil/^ 
and  aFO  (Fig.  223), 

aA\aO:\MA\FO. 
From  the  similar  triangles 
OAC  ".nd  Oac, 

aA'.aOwcC'.cO. 
Hence,  cC.cOw  MA  :  FO. 

I^t  OC=D{=MA),  Oc  =  d,  d.n^.FO=f  (the  focal  length 
of  the  lens).  Substituting  these  values  in  the  last  proportion,  we 
have  (d-D):d:'.D:/, 

d-D     D 


->:^-~^. 


Fig.  223. 


304 

Light 

Dividing  by  D, 

tf-D      I 

or 

I      I  _  I 

Laboratory  Exercise  j6 

PROBLEMS 

The  following  problems  will  help  to  familiarize  the  pupil  with  a  number 
of  facts  which  will  be  of  assistance  in  the  study  of  optical  instruments.  The 
problems  are  to  be  solved  by  means  of  the  formulas. 

1.  The  distance  of  the  image  is  how  many  times  the  focal  length  when 
the  distance  of  the  object  is  (</)  twice  the  focal  length?  (^)  ten  times  the 
focal  length?  (<-)  one  hundred  times?  (</)  one  thousand  times? 

2.  The  distance  of  the  image  is  how  many  times  the  focal  length  when 
the  distance  of  the  object  is  {a)  .9  the  focal  length?  (//)  .8?  {c)  .5?  (</)  .1? 

Do  the  distances  of  the  object  and  its  virtual  image  become  more  or  less 
nearly  equal  as  the  object  approaches  the  lens  from  the  principal  focus? 

3.  An  image  seen  in  a  simple  microscope  is  at  a  distance  of  10  in.  What 
is  the  distance  of  the  object  if  the  focal  length  of  the  lens  is  {a)  2  in.?  {h)  i 
in.?  (0  .5  in.? 

For  an  image  at  a  fixed  distance,  is  the  distance  of  the  ol)ject  more  or  less 
nearly  e(]ual  to  the  focal  length  as  the  focal  length  is  diminished? 

4.  What  is  the  per  cent  of  error  in  assuming  that  the  distance  of  the  ol>- 
ject  is  equal  to  the  focal  length  when  the  distance  of  the  image  is  10  in.  and 
the  focal  length  of  the  lens  (<i)  i  in.?  {b)  .5  in.? 

5.  An  object.  2  cm.  long  is  at  a  distance  of  50  cm.  from  a  lens  whose  focal 
length  is  15  cm.     Find  the  distance  of  the  image  and  its  length. 

6.  An  object  i  cm.  long  is^at  a  distance  of  1.7  cm.  from  a  lens  whose  focal 
length  is  2  cm.     find  the  distance  of  the  image  and  its  length. 

381.  Spherical  Aberration.  —  W'hen  light  from  any  object  is 
allowed  to  pass  through  only  the  central  portion  of  a  lens  (an 
opaque  screen  with  a  small  circular  ope#ing  being  used  for  this 
purpose),  the  image  is  formed  at  a  slightly  greater  distance  than 
it  is  when  formed  by  light  that  passes  through  the  lens  near  its 
edge  (the  light  through  the  central  portion  being  cut  off  by  an 
opaque  disk)    {Exp.). 

These  experiments  show  that  the  deviation  becomes  too  great 


Lenses 


305 


hlG.   224. 


toward  the  edge  of  the*  lens,  causing  convergence  of  this  light  to 
a  nearer  focus  than  that  of  the  light  passing  through  the  central 
portion  of  the  lens  (Fig.  224).  Hence,  when  the  whole  lens  is 
used,  all  the  light  cannot  be 
sharply  focused  on  a  screen 
at  the  same  time ;  and  the  ^ 
image  is  more  or  less  blurred 
by  the  light  that  is  out  of 
focus.  This  unequal  focusing  of  the  light  is  called  spherical 
aberration,  since  it  is  the  result  of  the  spherical  shape  of  the 
surfaces.  It  increases  toward  the  edge  of  the  lens;  and,  for 
lenses  of  equal  size,  it  is  greater,  the  less  the  focal  length.  Spher- 
ical aberration  is  corrected  in  the  large  lenses  of  astronomical 
telescopes  by  diminishing  the  curvature  of  the  surfaces  toward 
the  edge.  The  grinding  and  polishing  of  such  surfaces  is  very 
expensive,  as  it  must  be  done  by  hand,  hence  spherical  lenses 
are  used  in  the  smaller  telescopes  and  other  optical  instruments. 
Excellent  focusing  is  obtained  with  a  camera  by  using  a  dia- 
phragm with  a  small  opening,  which  admits  light  only  through  the 
central  portion  of  the  lens.  This  method  of  correction  greatly 
diminishes  the  intensity  of  the  light ;  but  the  intensity  is  still  suffi- 
cient for  the  purposes  of  photography. 

382.   The   Concave  Lens.  —  A   beam  of  light  parallel  to   the 
principal   axis  of  a  concave  lens  emerges  as  a  diverging   cone 

(Fig.  225)  {Exp.),  The 
vertex  F  of  this  cone  is 
the  apparent  source  of  the 
refracted  light,  and  is  called 
the  pnncipal focus.  It  is  a 
virtual  focus.  • 

Since  concave  lenses  al- 


Fio.  225. 


ways  increase  the  divergence  of  the  light  passing  through  them, 
the  images  formed  by  them  are  always  virtual  and  are  on  the 
same  side  of  the  lens  as  the  object.  These  images,  like  those 
formed  by  convex  mirrors,  are  always  erect,  and  smaller  than  the 


3o6  Light 


object,  and  are  nearer  the  lens  than  the'  object  is.  When  the 
object  is  moved  from  a  position  at  the  lens  to  an  indefinitely 
great  distance,  the  image  recedes  from  the  lens  to  the  principal 

focus.    These  characteristics 

^__ ^ :^f-^7"^^  of  the  image  are  explained 

by  figures  constructed  like 
Fig.  226.  In  locating  the 
virtual  conjugate  focus  of  any 
point,  we  make  use  of  two 
rays,  namely  :  ( i )  The  ray 
through  the  center  of  the  lens,  which  continues  in  the  same  direc- 
tion after  passing  through  the  lens;  and  (2)  the  ray  parallel  to 
the  principal  axis,  the  direction  of  which  after  emergence  is  from 
the  principal  focus. 

VI.  The  Eye 

383.  The  Structure  of  the  Eye.  —  The  human  eye  is  a  nearly 
spherical  ball  somewhat  less  than  an  inch  in  diameter.  Figure  227 
represents  the  top  view  of  a  horizontal  section  of  the  right  eye. 
The  outer  coat  or  wall  of  the  eye  is  thick  and  horny ;  it  gives  firm- 
ness to  the  eye  and  protects  the  delicate  parts  within.  Its  front  por- 
tion, the  cornea^  is  transparent ;  the  remainder  is  opaque  and  white. 

Behind  the  cornea  is  an  opaque  circular  diaphragm  called  the 
iris.  This  is  the  colored  part  of  the  eye,  and  the  circular  opening 
at  its  center  is  called  the  pupil.  The  iris  regulates  the  amount  of 
light  that  enters  the  eye  by  involuntary  muscular  action,  which 
causes  the  pupil  to  contract  when  the  intensity  of  the  light  in- 
creases and  to  expand  when  the  intensity  decreases.  The  iris 
also  corrects  spherical  aberration  by  permitting  light  to  pass 
through  only  the  central  portion  of  the  crystalline  lens,  which  is 
just  behind  it.  This  lens  is  a  double-convex,  transparent  solid 
made  up  of  concentric  layers  or  shells,  which  increase  in  density 
and  refractive  power  toward  the  center. 

The  cavity  between  the  cornea  and  the  crystalline  lens  is  filled 
with  a  watery  liquid  called  the  aqueous  humor.     The  large  cavity 


The  Eye 


307 


back  of  the  lens  is  filled  with  a  jelly-like  substance  called   the 
vitreous  humor. 

The   rear   half  of  the  eyeball  is  lined  with  a  semitransparent 
membrane,  called  the  retitia,  which  contains  a  network  of  nerve 

Ciliary  Muscle 


Optic  Nerve 

Fig.  227. 


Choroid 


fibers  branching  from  the  optic  nerve.  Light  incident  on  the 
retina  gives  rise  to  the  sensation  of  vision.  A  thin,  black  mem- 
brane (the  choroid  coat)  lies  between  the  retina  and  the  outer 
coat  of  the  eye,  and  extends  forward  to  the  iris,  with  which  it  is 
continuous.  This  membrane  absorbs  all  the  light  that  falls  upon 
it,  making  the  eye  a  dark  chamber. 

384.  The  Eye  as  an  Optical  Instrument.  — The  light  by  which 
we  see  passes  through  the  three  refractive  media  of  the  eye,  —  the 
aqueous  humor,  the  crystalline  lens,  and  the  vitreous  humor.  The 
index  of  refraction  of  the  aqueous  and  the  vitreous  humors  is 
1.337,  ^^^  ^^^^  average  for  the  lens  is  1.437.  Consequently  the 
successive  refractions  at  the  cornea  and  the  surfaces  of  the  lens 


308  Light 

together  make  the  light  strongly  convergent.  If  the  eye  is  perfect, 
the  convergence  is  just  sufficient  to  bring  the  light  to  a  focus  on 
the  retina,  which  thus  serves  as  a  screen  to  receive  the  real  and 
inverted  image  of  the  source  of  light.  How  the  formation  of  a  real 
image  upon  the  retina  results  in  sight  is  not  to  be  considered 
here  ;  it  is  a  question  to  which  no  complete  answer  can  be  given. 

"  The  inversion  of  images  in  the  eye  has  greatly  occupied  both 
physicists  and  physiologists,  and  many  theories  have  been  proposed 
to  explain  how  it  is  that  we  do  not  see  inverted  images  of  objects. 
The  chief  difficulty  seems  to  have  arisen  from  the  conception  of 
the  mind  or  brain  as  something  behind  the  eye  looking  into  it 
and  seeing  the  image  upon  the  retina ;  whereas  really  this  image 
simply  causes  a  stimulation  of  the  optic  nerve,  which  produces 
some  molecular  change  in  some  part  of  the  brain,  and  it  is  only 
of  this  change,  and  not  of  the  image,  as  such,  that  we  have  any 
consciousness.  The  mind  has  thus  no  direct  cognizance  of  the 
image  upon  the  retina,  nor  of  the  relative  positions  of  its  parts, 
and  sight  being  supplemented  by  touch  in  innumerable  cases,  it 
learns  from  the  first  to  associate  the  sensations  brought  about  by 
the  stimulation  of  the  retina  (although  due  to  an  inverted  image) 
with  the  correct  position  of  the  object  as  taught  by  touch."  — 
Ganot's  Physics. 

385.  The  Field  of  Distinct  Vision.  —  Although  you  can  see  the 
whole  of  both  of  these  pages  at  the  same  time,  you  will  find  that 
while  you  are  looking  steadily  at  one  word  you  cannot  read  more 
than  one  or  two  short  words  on  either  side  of  it.  While  the  field 
of  vision  (measured  by  the  angle  at  the  eye  within  which  objects 
can  be  seen  at  the  same  time)  is  very  large,  the  field  of  most 
distinct  vision  is  surprisingly  small,  as  is  shown  by  this  simple  test. 
We  are  seldom  conscious  of  the  fact,  however,  for  we  are  accus- 
tomed to  fix  the  attention  wholly  on  the  spot  at  which  we  are 
directly  looking.  In  looking  attentively  at  a  large  object,  a  rapid 
shifting  of  the  eyes  brings  successive  portions  of  it  into  distinct 
view. 

When  the  eyes  are  directed  toward  any  small  area,  its  image 


The  Eye 


309 


falls  upon  the  middle  of  the  back  part  of  the  retina.  This  is  the 
region  in  which  vision  is  most  distinct ;  it  is  called  the  yellow  spot 
(see  figure). 

386.  Adaptation  to  Different  Distances.  —  If  the  parts  of  the  eye 
were  all  fixed  in  shape  and  relative  position  and  the  image  of  a 
very  distant  object  were  exactly  focused  on  the  retina,  the  focusing 
would  remain  perfect  for  shorter  distances  down  to  about  twenty 
feet ;  but,  for  distances  less  than  this,  the  focus  would  be  beyond 
the  retina  J  causing  the  images  upon  the  retina  to  be  blurred,  and 
objects  would  appear  indistinct.  This  imperfection  of  focusing 
would  rapidly  increase  for  distances  under  three  feet. 

Since  we  are  able  to  see  both  near  and  distant  objects  distinctly, 
it  is  evident  that  the  eye  is  capable  of  some  form  of  adjustment 
for  distance.  This  adjustment  is  known  as  accointnodation.  It 
has  been  found  by  observations  upon  the  eye,  such  as  oculists 
are  able  to  make,  that  accommodation  is  effected  by  means  of 
the  crystalline  lens,  the  front  surface  of  which  moves  forward 
and  becomes  more  convex  when  near  objects  are  viewed.  This 
diminishes  the  focal  length  of  the  lens  and  increases  the  distance 


CI).IARY  MUSCLE 


FAR  NEAR 

Fig.  228. 


CIUARY  PROCESS 


of  its  center  from  the  retina,  both  of  which  changes  assist  in 
bringing  the  image  forward  to  the  retina.  The  left  half  of  Fig. 
228  represents  the  lens  adjusted  for  distant  vision  and  the  right 
half  for  near  vision. 

The  mechanism  of  accommodation  is,  in  its  main  features,  as 
follows  :  The  lens  is.  inclosed  in  a  membranous  sac,  round  the 
front  surface  of  which,  near  the  margin,  is  attached  a  membrane 


3IO  Light 

called  the  suspensory  ligament.  The  outer  edge  of  this  ligament 
is  attached  to  the  projecting  edge  of  a  circular  muscle  (the  ciliary 
muscle)  extending  round  the  eye  just  back  of  the  iris.  When  this 
muscle  is  relaxed,  the  suspensory  ligament  is  under  tension,  causing 
the  membrane  that  incloses  the  lens  to  press  upon  its  front  sur- 
face. The  lens  is  slightly  flattened  by  this  pressure,  and  thus 
brought  into  focus  for  distant  vision.  The  contraction  of  the 
muscle  draws  its  projecting  edge  forward.  This  relaxes  the 
suspensory  ligament,  relieving  the  pressure  from  the  front  surface 
of  the  lens,  which  moves  forward  and  becomes  more  convex. 

387.  The  Least  Distance  of  Distinct  Vision.  —  The  power 
of  accommodation  of  the  eye  is  limited.  When  the  object  is  at 
less  than  a  certain  distance,  the  effort  to  focus  the  eye  upon 
it  becomes  tiresome ;  and  within  a  somewhat  shorter  distance 
focusing  is  impossible.  These  distances  vary  considerably  with 
different  eyes ;  but  with  perfect  eyes  the  least  distance  of  distinct 
vision  is  about  25  cm.  (10  in.).  This  is  commonly  taken  as  the 
distance  of  most  distinct  vision  in  computing  the  magnifying  power 
of  optical  instruments. 

388.  Optical  Defects  of  the  Eye.  —  Short  Sight.  —  In  some  eyes 
the  image  of  a  distant  object  is  formed  in  front  of  the  retina,  the 
eyeball  being  too  long  or  the  curvature  of  the  cornea  or  the  lens 
too  great.  Such  eyes  are  said  to  be  near-sighted,  for  the  image 
falls  upon  the  retina  only  when  the  object  is  very  near.  This 
defect  is  corrected  by  wearing  concave  glasses,  which  offset  the 
excessive  convergence  within  the  eyes  by  increasing  the  divergence 
of  the  incident  light. 

Long  Sight.  —  In  some  cases  the  eye  is  too  short  or  the  crystal- 
line lens  not  sufficiently  converging,  and  the  focus,  even  for  distant 
objects,  would  fall  behind  the  retina  if  the  power  of  accommoda- 
tion were  not  exercised.  Such  eyes  are  far-sighted,  and  cannot 
be  focused  on  near  objects  without  a  fatiguing  effort,  if  at  all. 
Convex  glasses  correct  the  defect  by  supplementing  the  deficient 
convergence  within  the  eyes. 

In  old  age  the  crystalline  lens  loses  its  elasticity  and  becomes 


The  Eye 


311 


incapable  of  accommodation  for  near  vision.  Hence  old  people 
whose  eyes  were  perfect  in  earlier  years  see  distant  objects  dis- 
tinctly, but  require  convex  glasses  for  reading. 

Astigmatism.  —  This  defect  consists  in  unequal  curvature  of 
the  cornea  or  of  the  crystalline  lens  in  different  directions ;  as  a 
result  of  which  the  light  from  any 
point  of  an  object  does  not  con- 
verge to  a  point  on  the  retina, 
but  forms  a  line  instead.  An 
eye  that  is  astigmatic  cannot  be 
exactly  focused  for  vertical  and 
horizontal  lines  at  the  same  time  ; 
hence  Fig.  229  presents  a  simple 
test  for  this  defect.  The  radi.it- 
ing  lines  are  all  alike ;  but  to 
most  persons  they  will  appear 
unequally  distinct,  for  there  are 
very  few  eyes  that  are  not  astig- 
matic in  some  degree.  If  the 
eyes  are  strongly  astigmatic  (in- 
dicated   by    decidedly    unequal 


Fig.  229. 


distinctness  of  the  radiating  lines  in  the  figure),  glasses  should  be 
worn,  especially  for  reading.  To  correct  astigmatism,  one  surface 
of  the  lens  is  given  a  cylindrical  curvature  —  in  some  cases  concave, 
in  others  convex. 

389.  Binocular  Vision.  —  "  The  question  as  to  how  it  is  that  we 
see  objects  single  with  two  eyes  is  not  to  be  altogether  explained 
by  habit  and  association.  To  each  point  in  the  retina  of  one  eye 
there  is  a  corresponding  point,  similarly  situated,  in  the  other.  An 
impression  produced  on  one  of  these  points  is,  in  ordinary  circum- 
stances, undistinguishable  from  a  similar  impression  produced  on 
the  other,  and  when  both  at  once  are  similarly  impressed,  the 
effect  is  simply  more  intense  than  if  one  were  impressed  alone ; 
or,  in  other  words,  we  have  only  one  field  of  view  for  our  two 
eyes,  and  in  any  part  of  this  field  of  view  we  see  either  one  image, 


312  Light 

brighter  than  we  should  see  it  by  one  eye  alone,  or  else  we  see 
two  overlapping  images.  Tliis  latter  phenomenon  can  be  readily 
illustrated  by  holding  up  a  finger  between  one's  eyes  and  a  wall, 
and  looking  at  the  wall.  We  shall  see,  as  it  were,  two  transparent 
fingers  projected  on  the  wall.  One  of  these  transparent  fingers  is 
in  fact  seen  by  the  right  eye,  and  the  other  by  the  left,  but  our 
visual  sensations  do  not  directly  inform  us  which  of  them  is  seen 
by  the  right  eye,  and  which  by  the  left." —  Deschanel's  Natural 
Philosophy. 

It  is  principally  in  the  estimation  of  distances  and  in  the  per- 
ception of  relief  that  there  is  an  advantage  in  having  two  eyes. 

390.  The  Estimation  of  Distance.  —  The  line  through  the  cen- 
ters of  the  cornea  and  the  crystalline  lens  is  called  the  optic  axis 

jiaC\   of  the  eye.      When  we  look  at 
1 ■  ^s^  any  point,  the  optic  axes  of  both 

■ ^        i^IV    ^y^^  ^^^  directed  toward  it  (Fig. 

Fig.  230.  C^^y     230).      This   requires   a   greater 

or  less  convergence  of  the  axes  according  as  the  object  is  nearer 
or  farther  away ;  and,  in  the  case  of  near  objects,  the  amount 
of  the  convergence  is  the  principal  means  by  which  we  estimate 
the  distance. 

That  we  are  in  fact  less  able  to  judge  of  distances  with  one  eye 
than  with  both  is  easily  proved  by  trying  to  snap  a  cork  or  other 
small  object  from  the  back  of  a  chair  with  a  finger,  while  walking 
rapidly  past  the  chair  with  one  eye  shut.  The  object  will  almost 
invariably  be  missed,  because  the  eye  misjudges  the  distance. 

391.  The  Perception  of  Relief.  —  When  we  look  with  both  eyes 
at  any  object  that  is  not  very  distant,  the  point  of  it  to  which  the 
eyes  are  at  any  instant  directed  appears  single,  for  its  image  then 
falls  upon  corresponding  points  of  the  retinas.  At  the  same  time 
we  receive  a  somewhat  indistinct  impression  of  the  whole  object ; 
and  this  impression,  when  the  attention  is  directed  toward  it 
without  shifting  the  position  of  the  eyes^  is  found  to  involve  a  large 
amount  of  doubleness.  For  example,  if  we  look  at  a  long  pencil, 
held  in  the  hand  with  the  sharp  end  pointing  toward  the  chin, 


The  Eye 


•313 


a  b  c 

Fig.  231. 


it  will  appear  as  shown  in  «,  b,  or  c  of  Fig.  231,  according  as 

the  eyes  are  directed  toward  the  nearer  end, 

the  middle,  or  the   farther   end  of  it.     The 

impression   of  distinct   vision  for   the   point 

of  direct  observation  and  the  impressions  of 

indistinct  or  double  vision  for  the  other  points 

combine  to  give  us  the  perception  of  relief, 

or  of  form   in   three    dimensions.      With  a  single  eye,  relief  is 

only  imperfectly  perceived,  as  in  a  picture. 

392."  The  Principle  of  the  Stereoscope.  —  Since  the  two  eyes  see 
an  object  from  different  points  of  view,  its  image  on  the  retina  in 
one  eye  differs  slightly  from  that  in  the  other,  especially  when  the- 
object  is  near  ;  and  it  is  by  the  apparent  union  of  these  two  dis- 
similar views  that  we  see  the  object  in  rehef.  Hence  if  we  pre- 
sent to  each  eye  a  picture  of  an  object  as  it  appears  from  its  point 
of  view  and  direct  the  eyes  so  that  the  pictures  appear  to  be  super- 
posed, the  appearance  of  relief  will  be  perfectly  reproduced.  This 
is  illustrated  by  Fig.  232.     The  two  pictures,  A  and  B,  represent 


Fig.  232. 

the  tunnel  as  it  appears  to  the  left  eye  and  the  right  eye  respec- 
tively. Hold  the  book  so  that  one  picture  is  immediately  in  front 
of  each  eye,  and  place  a  card  between  the  pictures  and  perpen- 
dicular to  the  page,  so  that  each  eye  can  see  only  the  picture  on 
its  own  side.  Now  direct  the  eyes  as  if  you  were  looking  through 
the  book  at  a  point  some  distance  behind  it.  When  this  is  done, 
the  pictures  will  appear  to  move  together  and  unite  into  a  siijgle 


3H 


Light 


Fig.  233. 


view,  which  has  the  appearance  of  real  depth  extending  a  foot  or 
more  behind  the  book.  With  a  httle  practice,  the  card  between 
the  pictures  can  be  dispensed  with  ;  but  two 
additional  tunnels  will  then  be  indistinctly  seen, 
one  on  either  side,  the  one  on  the  left  side  being 
the  left  picture  as  seen  by  the  right  eye  and  the 
other  the  right  picture  as  seen  by  the  left  eye. 

The  stereoscope  is  an  instrument  designed  to 
aid  in  the  combination  of  slightly  dissimilar 
photographs  taken  with  a  double  camera,  or 
with  a  single  camera  from  two  positions  a  short 
distance  apart.  The  two  pictures,  A  and  B 
(Fig.  233),  are  mounted  on  the  same  card,  and 
are  viewed  through  the  half  lenses  M  and  N. 
The  magnified,  virtual  images  of  the  two  pictures  coincide  at  C. 
The  partition  P  prevents  each  eye  from  seeing  the  picture 
intended  for  the  other. 

393.  Visual  Angle  and  Angular  Size.  —  The  angle  between  the 
rays  that  enter  the  eye  from  the  extremities  of  an  object  is  called 
the  visual  angUy  or  the  angle  under  which  the  object  is  seen.  This 
angle  measures  the  angular  size  of  the  object.  Thus  in  Fig.  234 
the  angular  size  of  the  ob- 
ject AB  is  A  OB.  When  a 
small  coin  is  held  before  the 
eye  at  such  a  distance  that  it 
is  just  large  enough  to  con- 
ceal the  sun  from  that  eye, 
the  coin  at  that  distance  has  the  same  angular  size  as  the  sun. 
For  small  angles,  the  angular  size  of  an  object  is  proportional  to 
its  actual  size  and  inversely  proportional  to  its  distance  from  the 
observer. 

The  size  of  the  image  on  the  retina  is  proportional  to  the  angu- 
lar size  of  the  object,  and  hence  increases  as  the  distance  of  the 
object  decreases  (see  figure).  This  explains  why  an  object  is 
seen  more  and  more  distinctly  as  it  is  brought  toward  the  eye, 


Fig.  234. 


Optical  Instruments 


315 


until  the  least  distance  of  distinct  vision  is  reached.  No  more 
detail  of  the  form  and  features  of  a  man  could  be  distinguished  at 
a  distance  of  100  yards  than  could  be  seen  in  a  photograph  of 
him  half  an  inch  long  when  held  at  arm's  length ;  for  the  man  and 
the  photograph,  at  these  distances,  would  have  the  same  angular  size. 
When  the  size  of  an  object  is  approximately  known,  as  the 
height  of  a  tr6e  or  a  house,  its  angular  size  serves  as  a  basis  for 
estimating  its  distance.  The  various  forms  of  telescopes  and 
microscopes  serve  the  purpose  of  increasing  the  visual  angle. 


VII.    Optical  Instruments 

394.  The  Simple  Microscope.  —  A  converging  lens,  when  used 
as  a  simple  microscope,  forms  a  magnified,  virtual  image  of  the 
object,  as  explained  in  Art.  379.  Lenses  for  such  use  are  mounted 
in  frames  and  supports  of  various  forms;  but  these  are  merely 
matters  of  convenience  and  need  not  be  considered. 

It  can  be  shown  that  the  magnification  produced  by  a  simple 
microscope  is  greatest  when  the  eye  is  placed  close  to  the  lens 
and  the  distance  of  the  object  is  such  that  the  image  is  at  the 
least  distance  of  distinct  vision  (Art.  387).     This  maximum  mag- 


ax- 


Fig.  235. 


nification  is  called  the  magnifying  power  of  the  miscroscope,  and 
may  be  defined  as  the  ratio  of  the  angular  size  of  the  image  to  the 
angular  size  of  the  object,  both  being  at  the  least  distance  of  dis- 
tinct vision  (angle  aOb :  angle  AOB\  Fig.  235),  or  as  the  ratig 


3i6 


Light 


of  the  length  of  the  image  at  this  distance  to  the  length  of  the 
object  {ab  :  A'B'  or  ab  :  AB). 

From  similar  triangles,  ab  :  AB : :  EO  :  DO  \  hence  the  magni- 
fying power  is  equal  to  the  ratio  of  the  least  distance  of  distinct 
vision,  EOy  to  the  distance  of  the  object,  DO. 

If  the  focal  length  of  the  lens  does  not  exceed  5  cm.  (and  it 
is  generally  less),  it  is  only  slightly  greater  than  DO  and  may  be 
substituted  for  it.  Hence  if  L  denote  the  least  distance  of  dis- 
tinct vision  and  /  the  focal  length  of  the  lens,  we  obtain  the  con- 
venient ratio  Z  :  /  as  the  approximate  measure  of  the  magnifying 
power  of  a  simple  microscope.     L  is  taken  as  25  cm.  or  10  inches. 

Laboratory  Exercisf  57. 

395.  The  Compound  Microscope.  —  The  images  formed  by  a 
single  lens  are  imperfect  if  the  magnification  is  very  great ;  hence 
compound  microscopes  are  used  for  observing  very  minute  objects. 
A  compound  microscope  in  its  simplest  form  consists  of  two  lenses 
both  of  short  focal  length,  called  respectively  the  objective,  O,  and 
the  eyepUce^  E  (Fig.  236).     The  objective  is  placed  at  a  distance 


Fig.  236. 


only  slighdy  greater  than  its  focal  length  from  the  object,  AB ; 
and  a  magnified,  real  image,  ab,  is  formed  at  a  much  greater  dis- 
tance on  the  other  side  of  the  lens.  The  eyepiece  is  placed  at  a 
distance  a  little  less  than  its  focal  length  from  the  real  image, 
and  forms  a  virtual  image  of  it,  a'b'.     The  eyepiece  serves  as  a 


Optical  Instruments  317 

simple    microscope   for  viewing  the  real   image  formed  by  the 
objective. 

The  magnification  produced  by  the  objective  is  measured  by 
the  ratio  of  the  distance  of  the  real  image  from  the  objective  to 
the  distance  of  the  object  from  it  (Art.  378).  Let  ^denote  the 
distance  of  the  image  and  F  the  focal  length  of  the  objective. 
The  latter  may  be  taken  as  the  distance  of  the  object  with  only 
a  slight  error;  hence  the  magnifying  power  of  the  objective  is 
approximately  d  -^  F,  and  is  inversely  proportional  to  its  focal 
length.  The  magnifying  power  of  the  eyepiece  is  L  -5-/,  /  being 
its  focal  length,  and  Z  the  least  distance  of  distinct  vision.  The 
magnifying  power  of  the  microscope  is  the  product  of  the  magni- 
fications produced  separately  by  the  objective  and  the  eyepiece, 
or  approximately  {d  -i-  F)x(L  -f-/). 

If  the  objective  and  eyepiece  consisted  of  only  a  single  lens 
each,  as  we  have  assumed,  the  image  would  be  distorted,  indis- 
tinct, and  bordered  with  the  colors  of  the 
rainbow.  The  coloring  is  explained  in  Art. 
414;  it  is  avoided  by  using  one  or  more 
achromatic  lenses  (Art.  415)  for  the  object- 
ive and  two  convex  lenses,  placed  some  dis- 
tance apart,  for  the  eyepiece.  The  distances 
between  the  lenses  of  both  objective  and 
eyepiece  and  the  curvatures  of  their  surfaces 
are  so  chosen  as  to  correct  all  defects  of  the 
image. 

Figure  237  represents  a  compound 
microscope  mounted  on  a  stand.  The 
objective  and  eyepiece  are  at  the  ends 
of  the  tube.     The  object  is  placed  on  _ 

the  stage  and  is  brought  into  focus  by 
raising  or  lowering  the  tube  by  means  of  a  thumb  screw  at  the  side. 

Laboratory  Exercise  62. 

396.  The  Astronomical  Telescope.  —  The  astronomical  tele- 
scope, like  the  compound  microscope,  consists  essentially  of  two 


3i8 


Light 


lenses,  an  objective,  O^  and  an  eyepiece,  E  (Fig.  238).  An 
inverted  image,  ab^  of  a  distant  object  is  formed  by  the  objective 
at  its  principal  focus.  The  parallel  rays  JK  and  NO  come  from 
a  point  on  the  lower  side  of  the  object,  and  converge  to  the  cor- 


FlG.  238. 

responding  point  b  in  the  image ;  similarly  the  parallel  rays  MO 
and  HI  from  a  point  at  the  top  of  the  object  converge  to  a.  The 
eyepiece  forms  a  magnified,  virtual  image,  a^d\  of  the  real  image, 
as  in  the  compound  microscope.  Both  the  real  and  the  virtual 
images  are  inverted  with  respect  to  the  object. 

The  size  of  the  image  formed  by  the  objective  is  proportional  to 
its  focal  length  (4,  Art.  378);  and,  since  any  increase  in  the  size 
of  the  real  image  would  result  in  a  proportional  increase  in  the 
size  of  the  virtual  image  formed  by  the  eyepiece,  it  follows  that 
the  magnifying  power  of  the  telescope  is  proportional  to  the  focal 
length  of  the  objective.  It  is  for  this  reason  that  powerful  tele- 
scopes are  long.  The  eyepiece  is  always  of  short  focal  length,  and 
increases  the  magnification.  The  magnifying  power  of  a  telescope 
is  defined  as  the  ratio  of  the  visual  angle  when  an  object  is  viewed 
through  it  (angle  a'Eb'  or  aEb)  to  the  visual  angle  with  the 
naked  eye  (angle  MON  ox  its  equal  aOb^.  Since  angles  aEb 
and  aOb  are  subtended  by  the  same  line  ab,  they  are  (for  small 
angles)  inversely  proportional  to  the  distances  of  their  vertices 
from  this  line  ;  i.e.  angle  aEb :  angle  aOb  =  OC:  CE.  Let  F 
denote  the  focal  length  of  the  objective,  OC-,  and  /  the  focal 
length  of  the  eyepiece,  which  is  (very  nearly)  equal  to  CE.  Sub- 
stituting in  the  proportion,  we  have  angle  aEb  :  angle  aOb  =  F:/ 


Optical  Instruments 


319 


(approximately)  ;  that  is,  the  magnifying  power  is  measured  by 
the  ratio  of  the  focal  length  of  the  objective  to  the  focal  length  of 
the  eyepiece. 

The  objective  is  of  large  diameter  in  order  to  receive  as  much 
light  as  possible  from  the  heavenly  body  under  observation.  This 
increases  the  brightness  of  the  image,  and  renders  visible  count- 
less stars  that  are  never  seen  with  the  unaided  eye.  Even  an 
opera  glass  greatly  increases  the  number  of  visible  stars.  The 
objective  consists  of  an  achromatic  pair  of  lenses,  which,  in  large 
instruments,  are  placed  several  inches  apart.  To  secure  the 
greatest  possible  perfection,  large  objectives  are  ground  and 
polished  by  hand  (Art.  381).  The  eyepiece  is  like  that  of  the 
compound  microscope. 

Laboratory  Exercise  60. 

397.  The  Terrestrial  Telescope.  —  The  fact  that  heavenly 
bodies  appear  inverted  through  an  astronomical  telescope  is 
unimportant ;  but  for  the  observation  of  terrestrial  objects  the 
image  must  be  erect.     An  additional  convex  lens,  Z  (Fig.  239),  is 


b" 

Fig.  239. 

sufficient  to  reinvert  the  image.  The  inverted  image  ad  is  formed 
by  the  objective  (not  shown  in  the  figure).  If  the  lens  Z  is  at 
twice  its  focal  length  from  this  image,  it  will  form  a  real  and  erect 
image  a'b'  at  an  equal  distance  on  the  other  side.  The  eyepiece 
forms  an'  erect  image  of  a'b'.     Telescopes  are  constructed  with 


320 


Light 


two  lenses  to  reinvert  the  image,  since  better  resnlts  are  obtained 
than  with  a  single  lens.  A  small  terrestrial  telescope  is  called  a 
spyg/ass. 

398.  The  Galilean  Telescope.  —  This  telescope  receives  its  name 
from  Galileo,  who,  if  not  the  first,  was  among  the  first  to  constnict 
such  an  instrument.  It  is  the  earliest  form  of  telescope,  and  also 
the  simplest,  since  it  gives  erect  images  with  only  two  lenses.    The 


Fir..  240. 

objective,  O  (Fig.  240),  is  a  convex  lens  as  in  the  astronomical 
telescope,  and  would  form  a  real  and  inverted  image,  ab,  at  its 
principal  focus  if  the  light  were  not  intercepted  by  the  eyepiece,  E, 
This  is  a  concave  lens,  and  is  placed  at  a  distance  equal  to  or  only 
very  slightly  exceeding  its  own  focal  length  from  the  focus  of  the 
objective. 

The  light  incident  upon  the  eyepiece  from  any  point  of  the 
object  IS  convergent ;  after  passing  through  the  eyepiece  it  is  very 
slightly  divergent  Thus  the  parallel  .rays  JK  and  NO  from  a 
point  at  the  bottom  of  the  object  would  converge  to  the  point  b  of 
the  real  image  if  it  were  not  for  the  refraction  produced  by  the 
eyepiece ;  but,  as  a  result  of  this  refraction,  they  appear  to  come 
from  b\  All  the  light  transmitted  by  the  eyepiece  is  similarly 
refracted,  forming  the  erect,    virtual  image  a^b'. 

The  angular  size  of  the  image  is  a'Eb'  or  its  equal  aEbj  and 
the  angular  size  of  the  object  MON or  its  equal  aOb.  But  for 
small  angles,  angle  aEb  :  angle  aOb  =  OC:  EC  =F  :  /,  i^ being 
the  focal  length  of  the  objective  and  /  that  of  the  eyepiece. 
Hence  the  magnifying  power  is  measured  by  the  ratio  of  the  focal 


Optical   Instruments 


32 


length  of  the  objective  to  the  focal  length  of  the  eyepiece,  as  in  the 
astronomical  telescope.  For  equal  power  the  Galilean  telescope 
is  the  shorter  of  the  two  by  twice  the  focal  length  of  the  eyepiece. 
(Why?)     It  has  a  still  greater  advantage  in  this  respect  over  the 


Fig.  341. 

terrestrial  telescope,  since  the  lens  in  the  latter  for  erecting  the 
image  increases  the  length  without  increasing  the  power. 

An  opera  glass  or  a  field  glass  is  a  pair  of  Galilean  telescopes 
mounted  together,  one  for  each  eye.  The  magnityin^  power  of  the 
opera  glass  is  from  two  to  three,  that  of  the  field  glass  is  from  four 
to  eight. 

Laboratory  Exercise  61. 

399.  The  Optical  Lantern.  —  This  is  an  instrument  by  means 
of  which  highly  magnified  images  of  transparent  photographs  and 
drawings  are  projected  upowa 
white  screen  in  a  dark  room. 
Its  essential  parts  are  a  strong 
source  of  light,  A  (Fig.  241); 
a  condensing  lens,  L  (gen- 
erally double),  for  concen- 
trating the  light  upon  the 
picture  or  "slide,"  F]  and 
an  achromatic  objective,  L\ 
placed  at  a  little  more  than 
its  focal  length  from  the  slide. 
The  image  is  formed  at  a  relatively  great  distance,  and  is  corre- 


FlG.  242. 


322  Light 


spondingly  magnified.     In  order  that  it  may  be  erect,  the  slide  is 
inverted. 

400.  The  Photographer's  Camera.  —  The  tube  AD  of  this  instru- 
ment contains  an  achromatic  convex  lens,  which  forms  an  inverted 
image  upon  a  ground-glass  screen  at  C.  The  image  is  focused  by 
moving  either  the  part  of  the  tube  that  carries  the  lens  or  the  glass 
screen.  When  the  focus  has  been  obtained,  the  screen  is  replaced 
by  a  plate  holder  containing  a  sensitized  glass  plate  (Fig.  242). 

PROBLEMS 

1.  ^^^lat  U  the  magnifying  power  of  a  simple  microscope  whose  focal 
length  is  (a)  2  in.?  (//)  .5  in.? 

2.  Show  from  the  formula  of  Art.  394  that  the  magnifying  power  of  a 
simple  microscope  is  inversely  proportional  to  its  focal  length.  Draw  figures 
to  illustrate. 

3.  Assuming  that  the  real  image  is  formed  5  in.  from  the  objective,  find 
the  magnifying  power  of  a  compound  microscope  (a)  with  a  .5  in.  objective 
and  a  2  in.  eyepiece  ;    {b)  with  a  \  in.  objective  and  a  ^  in.  eyepiece. 

4.  The  great  telescope  of  the  Lick  Observatory  is  57  ft.  long.  What  is  its 
magnifying  power  (a)  when  fitted  with  a  2  in.  eyepiece?  {b)  when  fitted  with 
a  \  in.  eyepiece? 

5.  The  objective  of  a  field  glass  has  a  focal  length  of  7.5  in.,  and  the  eye- 
piece a  focal  length  of  1.25  in.  Find  the  length  of  the  instrument  and  its 
magnifying  power. 

6.  How  is  the  size  of  a  photograph  affected  {a)  by  the  distance  of  the 

object  from  the  camera?  (^)  by  the  focal  length  of  the  lens  of  the  camera? 

». 

YIU.  Dispersion  and  Color 

401.  Decomposition  and  Recomposition  of  White  Light ;  Dis- 
persion.—  Sunlight  is  white  ;  but  after  refraction  through  uncolored, 

transparent  media,  it  is  often  highly 
colored.  This  effect  of  refraction  is 
strikingly  exhibited  when  a  beam  of 
sunlight  is  admitted  into  a  darkened 
room  through  a  long,  narrow  slit,  and 
Fig.  243.  ggj^^  through  a  prism  whose  refracting 

edge  is  parallel  to  the  slit.     If  the  slit  and  prism  are  vertical  and 


Dispersion  and  Color  323 

the  path  of  the  beam  horizontal,  Fig.  243  will  represent  a  hori- 
zontal section.  When  the  refracted  light  is  caught  upon  a  white 
screen  at  a  distance  of  a  few  meters  from  the  prism,  it  appears  as 
a  horizontal  band  of  light,  VJ?,  rounded  at  the  ends  and  colored 
in  all  the  tints  of  the  rainbow.  This  band  of  light  is  called  the 
so/ar  spectrum.  It  consists  of  an  indefinitely  great  number  of 
colors,  which  blend  imperceptibly  into  each  other.  The  seven 
principail  colors,  named  in  order,  are  violet^  indigo,  blue,  greeriy 
yellow,  orange,  and  red.  (The  paths  of  only  the  red  and  the  violet 
light  are  shown  in  the  figure.)  The  deviation  is  greatest  for  the 
violet  light  and  least  for  the  red  {Exp.). 

When  a  convex  lens  is  placed  in  the  path  of  the  refracted  light 
in  such  a  position  that  the  prism  and 
the  screen  are  at  conjugate  focal  dis- 
tances (Fig.  244),  the  lens  converges 
the  different  colors  to  the  same  spot 
upon  the  screen,  and  this  spot  is  white  .- 
{Exp.).  The  prism,  by  unequal  re- 
fraction or  dispersion,   separates  the 

light  of  different  colors,  all  of  which  are  present  in  the  incident 
beam  of  white  light.  When  these  colors  are  reunited  by  the  lens, 
they  again  form  white  light. 

When  the  prism  and  lens  are  removed  and  the  screen  alone  is 
placed  in  the  path  of  the  beam,  an  elongated  and  imperfect  image 
of  the  sun  is  formed  upon  it.  (When  nearly  the  whole  length  of 
the  slit  is  covered,  the  image  of  the  sun  is  round.)  This  image  is 
white,  since  it  is  composed  of  all  the  colors  of  the  incident  light. 
When  the  prism  is  interposed,  the  unequal  deviation  of  the  light 
of  different  colors  spreads  the  colored  images  of  the  sun  out  in 
a  line,  forming  the  spectrum.  Since  the  number  of  these  images 
is  unlimited,  there  is  much  overlapping  and  mixing  of  the  colors, 
and  the  spectrum  is  said  to  be  impure. 

402.  Production  of  a  Pure  Spectrum ;  Fraunhofer's  Lines.  — 
When  a  convex  lens  is  placed  in  the  path  of  the  beam  at  conjugate 
focal  distances  from  the  slit  and  the  screen,  the  distance  from  the 


324  Light 

slit  being  the  shorter,  an  enlarged  white  image  of  the  slit  is  formed 

upon  the  screen.*     When  the  prism 

is  placed  near  the  lens  (Fig.  245),  it 

produces  dispersion  of  the  colors,  as 

before  ;  but  the  Uns  brings  each  color 

to  a  focus  upon  the  screen^  forming 

Fig.  245.  ^  spectrum  which  consists  of  a  series 

of  narrow  and  only  very  slightly  overlapping  colored  images  of  the 

silt.     This  spectrum  is  rectangular,  and  is  much  wider  than  that 

produced  without  the  lens  {Exp.). 

Narrowing  the  slit  diminishes  the  overlapping  of  the  images,  and 
hence  makes  the  spectrum  more  nearly  pure ;  but  its  brightness  is 
correspondingly  diminished.  When  the  slit  is  very  narrow  and  the 
lens  exactly  focused,  the  spectrum  is  crossed  by  many  dark  lines 
at  right  angles  to  its  length  (/>.  in  a  direction  parallel  to  the  slit). 
These  lines  are  called  Fraunho/er's  iines^  after  the  celebrated 
optician  of  Munich,  who  first  studied  and  gave  a  detailed  descrip- 
tion of  them.  They  represent  missing  images  of  the  s/it,  and 
indicate  that  light  of  certain  colors  is  absent  from  sunlight.  When 
the  width  of  the  slit  is  only  very  slightly  increased,  the  overlapping 
of  the  images  on  each  side  of  the  dark  lines  obliterates  them. 

403.  Virtual  Spectrum.  —  If  the  preceding  experiments  cannot 
be  performed  for  want  of  a  dark  room  or  a  porte  lumiere  for 
directing  a  beam  of  sunlight  into  the  room,  the  observation  of  a 
virtual  spectrum  will  serve  as  a  substitute.  The  experiment  consists 
in  looking  through  a  prism  at  a  slit  about  a  millimeter  wide  in  a 
piece  of  black  cardboard.  The  cardboard  should  be  held  up  at 
arm*s  length  before  a  window,  with  the  sky  for  a  background  and 
with  the  slit  horizontal.  The  prism  is  held  close  to  the  eyes,  with 
its  edges  parallel  to  the  slit.     If  the  refracting  edge  of  the  prism 

1  It  b  only  the  light  from  the  same  portion  of  the  sun's  disk  that  is  parallel.  The 
rays  in  a  sunbeam  that  come  from  opposite  sides  of  the  sun  are  at  an  angle  of  about 
half  a  degree  {AOB,  Fig.  245).  The  lens,  in  the  position  described,  brings  all  the 
light  that  passes  through  any  point  of  the  slit  to  the  same  point  of  the  image,  and 
hence  forms  an  image  of  the  slit. 


Dispersion  and  Color 


325 


^ 


^;:- 


Fig.  246. 


is  on  the  lower  side,  the  cardboard  will  appear  at  an  angle  of  about 
40°  below  its  true  position  (Fig.  246) ;  and  the  observer,  looking 
obliquely  downward  at  this  angle,  will  see  a 
virtual  spectrum  consisting  of  a  series  of  over- 
lapping colored  images  of  the  slit.  Since 
the  violet  light  is  refracted  the  most,  the  vio- 
let image  of  the  slit  will  be  the  lowest  (see 
figure). 

Laboratory  Exercise  58, 
404.   The  Nature  of  Color.  —  When  any  part  of  the  light  com- 
posing a  spectrum  is  allowed  to  pass  through  a  second  prism,  it  is 
again  refracted,  but  its  color  remains  unchanged  (Fig.  247).     The 

colors  of  the  spec- 
^  trum   are  simple  or 

elementary;  that  is, 
they  cannot  be  fur- 
ther decomposed 
{Exp.). 

It  is  found  by  ex- 
periment ^  that  the 
wave  lengths  of  the  colors  of  the  spectrum  increase  regularly  from 
the  violet  to  the  red ;  and  that  the  same  elementary  tolor  always 
has  the  same  wave  length  (in  the  same  medium)  whether  its 
source  is  the  sun  or  any  other  luminous  body.  It  follows  that  the 
physical  cause  of  the  sensation  of  color  is  nothing  else  than  the 
wave  length  of  the  light ;  in  other  words,  the  sensation  produced 
by  an  elementary  color  is  determined  by  its  frequency  {i.e.  the 
number  of  waves  that  pass  any  point  in  a  second),  just  as  the  pitch 
of  a  sound  is  determined  by  the  vibration  number  of  the  sounding 
body  or  the  frequency  of  the  sound  waves. 

As  the  temperature  of  a  body  rises,  the  vibration  of  its  mole- 
cules becomes  more  and  more  rapid,  and  shorter  waves  are  set  up 
in  the  ether.     At  about  525°  C.  (Art.  331)  some  of  the  molecules 

1  It  is  beyond  the  scope  of  elementary  physics  to  discuss  the  methods  by  which 
the  wave  lengths  of  light  are  measured. 


Fig.  247. 


326  Light 

vibrate  with  sufficient  rapidity  to  give  out  red  light.  As  the  tem- 
perature continues  to  rise,  additional  colors  are  given  out  in  order 
from  red  to  violet,  and  the  color  of  the  body  changes  from  red 
through  orange  and  yellow  to  white. 

"  That  which  we  call  white  light  is,  in  the  state  in  which  we 
receive  it  from  such  a  body  as  a  white-hot  bar  of  iron,  or  perhaps 
in  its  purest  form  from  the  crater  of  the  positive  pole  of  the  electric 
arc,  a  mixture  of  long  and  short  waves ;  waves  of  all  periods 
within  the  range  of  visibility  are  either  continuously  present  or, 
if  absent  for  a  time,  are  absent  in  such  feeble  proportions  or  for 
such  short  intervals  that  they  are  not  appreciably  missed  by  the 
eye.  White  light  of  this  kind  is  comparable  to  an  utterly  discord- 
ant chaos  of  sound  of  every  audible  pitch  ;  such  a  noise  would 
produce  no  distinct  impression  of  pitch  ;  and  so  white  light  is 
uncolored."  —  Daniell's  Principles  of  Physics. 

405.  The  Invisible  Spectrum.  — The  luminous  ether  waves  vary 
in  length  from  .0000767  cm.  for  the  extreme  red  to  .0000397  cm. 
for  the  extreme  violet.  The  interval  between  these  extremes, 
expressed  as  in  music,  is  somewhat  less  than  one  octave.  Thus 
we  see  that  the  range  of  sensibility  of  the  eye  is  much  less  than 
that  of  the  ear  (Art.  304).  That  there  are  many  octaves  of  non- 
luminous  ether  waves  extending  both  above  and  below  the  visible 
portion  of  the  spectrum  is  proved  by  their  chemical,  heating,  and 
electrical  effects. 

406.  Cause  of  Dispersion.  —  \Vhenever  light  is  refracted,  the 
elementary  colors  of  which  it  is  composed  are  refracted  unequally, 
although  in  many  cases  the  dispersion  is  not  sufficient  to  be 
noticeable.  This  unequal  deviation  indicates  that  the  index  of 
refraction  of  a  substance  varies  with  the  color  of  the  incident 
light;  and,  since  the  index  of  refraction  is  the  ratio  of  the  velocity 
of  light  in  a  vacuum  to  its  velocity  in  the  substance  (Art.  363),  it 
is  evident  that  lights  of  different  colors  {or  ether  waves  of  different 
lengths^  travel  with  unequal  velocities  in  the  same  substance. 
This,  therefore,  is  the  cause  of  dispersion. 

With  few  exceptions,  the  deviation  increases  continuously  from 


Dispersion  and  Color  327 

the  red  to  the  violet,  as  in  the  preceding  experiments,  the  velocity 
of  the  shorter  waves  being  less  than  that  of  the  longer  in  most 
substances. 

407.  Color  of  Opaque  Bodies.  —  When  a  very  narrow  strip  of 
white  paper,  pasted  on  a  piece  of  black  cardboard,  is  viewed 
through  a  prism  as  the  slit  was  in  the  experiment  of  Art.  403,  the 
light  from  it  is  resolved  into  a  complete  spectrum,  the  colors  of 
which  have  the  same  relative  intensity  as  in  the  spectrum  of  direct 
sunlight.  Any  opaque  body  which,  like  the  white  paper,  reflects 
all  the  elementary  colors  of  the  incident  light  in  equal  proportions 
appears  white  when  white  light  falls  upon  it;  but  when  the  inci- 
dent light  is  colored,  the  body  appears  of  the  same  color.  Thus 
when  a  spectrum  is  thrown  upon  a  white  screen,  the  part  of  the 
surface  upon  which  the  red  light  falls  appears  red,  the  part  upon 
which  the  blue  light  falls  appears  blue,  etc. ;  for  each  part  reflects 
the  color  that  it  receives. 

When  a  narrow  strip  of  colored  paper  is  viewed  through  a  prism, 
the  light  from  it  is  resolved  into  an  incomplete  spectrum  ;  gener- 
ally half  or  more  of  the  spectrum  is  either  wanting  or  very  faint. 
The  spectrum  of  a  blue  strip,  for  example,  will  probably  be  found 
to  consist  of  violet,  indigo,  blue,  and  green  ;  that  of  a  yellow  strip, 
of  green,  yellow,  orange,  and  some  red.  A  similar  analysis  of  the 
light  from  different  colored  bodies  shows  that,  with  few  exceptions, 
the  light  reflected  by  them  is  composite^  i.e.  it  is  composed  of  a 
number  of  elementary  colors.  Any  body  that  reflects  some  of 
the  elementary  colors  of  white  light  in  larger  proportion  than  it 
does  others  is  colored,  its  color  being  determined  by  the  combined 
effect  of  all  the  colors  that  it  reflects.  A  body  reflecting  no  light 
would  be  perfectly  black.  White,  black,  and  the  diff"erent  shades 
of  gray  differ  only  in  brightness  ;  each  reflects  the  different  elemen- 
tary colors  in  the  same  proportions  in  which  it  receives  them. 

Color,  regarded  as  a  property  of  opaque  bodies,  is  therefore 
merely  the  power  of  reflecting  light  of  certain  wave  lengths  either 
exclusively  or  in  larger  proportions  than  others,  the  light  that  is 
not  reflected  being  absorbed.     It  is  further  evident  that  bodies 


328  Light 

have  no  color  of  their  own.  A  white  body,  as  stated  above,  takes 
the  color  of  the  incident  light ;  a  colored  body  appears  of  its  natu- 
ral color  only  when  the  incident  light  cqntains  all  the  elementary 
colors  that  it  is  capable  of  reflecting.  This  is  strikingly  illustrated 
by  holding  colored  papers  in  different  parts  of  the  solar  spectrum 
thrown  upon  a  screen  in  a  darkened  room.  A  piece  of  green 
paper,  for  example,  will  appear  black  in  the  violet,  indigo,  orange, 
or  red,  being  incapablfe  of  reflecting  these  colors  ;  in  the  blue  it 
will  probably  appear  a  dark  blue,  and  in  the  yellow  a  dirty  yellow, 
due  to  the  reflection  of  a  little  of  these  colors ;  in  the  green  it  will 
appear  at  least  very  nearly  of  its  natural  color  {Exp.). 

This  explains  why  some  bodies  do  not  appear  of  the  same  color 
by  artificial  light  as  by  daylight.  Most  artificial  lights  are  deficient 
in  violet  and  blue,  and  hence  are  more  or  less  yellowish.  In 
such  a  light,  pale  yellow  is  scarcely  distinguishable  from  white, 
and  blue  is  often  mistaken  for  green.  The  greenish  appearance 
of  blue  is  due  to  the  fact  that  blue  pigments  reflect  violet  and 
green  as  well  as  blue  light,  and  green  predominates  in  the  light 
that  they  reflect  when  illuminated  by  light  that  contains  little 
violet  and  blue. 

408.  Color  of  Transparent  Bodies.  —  A  transparent  body  is 
colored  if  it  is  more  transparent  to  some  of  the  colors  of  white 
light  than  to  others,  its  color  being  that  which  results  from  the 
J  mixture  of  all  the  transmitted  colors.  The  remaining  colors  of 
the  incident  light  are  absorbed  on  the  way  through  the  medium. 
This  action  of  a  colored  medium  is  called  selective  absorption.  If 
one  or  more  of  the  elementary  colors  that  a  body  can  transmit  are. 
not  present  in  the  incident  light,  the  body  will  not  appear  of  its 
natural  color ;  and  if  none  are  present,  it  will  appear  opaque,  since 
in  this  case  no  light  will  be  transmitted. 

The  light  transmitted  by  a  transparent  body,  as  a  piece  of  glass, 
may  be  analyzed  by  observing  either  its  real  or  virtual  spectrum. 
To  obtain  the  latter,  a  slit  is  observed  through  a  prism,  as  de- 
scribed in  Art.  403,  with  one  end  of  the  slit  covered  by  the  trans- 
parent body.     This  gives  a  complete  spectrum  from  the  uncovered 


Dispersion  and  Color  329 

end  of  the  slit,  and,  beside  it,  the  spectrum  of  the  light  transmitted 
through  the  body  (Lab.  Ex.).  To  obtain  the  real  spectrum,  a 
solar  spectrum  is  projected  upon  a  screen  in  a  darkened  room  by 
means  of  a  slit,  lens,  and  prism,  as  described  in  Art.  402  ;  and  the 
body  is  held  so  as  to  cover  either  the  upper  or  lower  end  of  the 
slit.  The  solar  spectrum  and  the  spectrum  of  the  transmitted 
light  will  then  be  projected  upon  the  screen,  one  above  the  other, 
making  a  comparison  of  the  two  very  easy. 

It  will  be  found  by  either  of  these  methods  of  observation  that 
the  light  transmitted  by  colored  bodies  is  composite  and,  with 
few  exceptions,  gives  a  considerable  portion  of  the  complete 
spectrum.  Blue  (cobalt)  glass  transmits  violet,  blue,  green,  and 
some  red  ;  yellow  glass  transmits  red,  orange,  yellow,  and  green ; 
red  (ruby)  glass  transmits  red  and  a  little  orange  (Exp.). 

When  two  transparent  bodies  of  different  color  are  placed  before 
the  slit,  one  in  front  of  the  other,  the  light  that  passes  through 
both  undergoes  a  double  process  of  selective  absorption ;  and  its 
spectrum  therefore  consists  only  of  the  color  or  colors  that  are 
common  to  the  light  transmitted  by  the  two  separately.  Thus 
green  is  the  only  one  of  the  colors  transmitted  by  either  blue  or 
yellow  glass  that  is  also  transmitted  by  the  other ;  hence  the  two 
together  appear  green.  Similarly,  the  combination  of  red  and 
blue,  red  an^  green,  or  orange  and  blue  glass  is  very  nearly 
opaque,  since  no  color  that  they  separately  transmit  in  considera* 
ble  quantity  is  common  to  both  {Exp.). 

409.  Color  of  Bodies  containing  Suspended  Particles.  —  A  gas 
or  a  liquid  which,  of  itself,  is  colorless  becomes  colored  when 
it  contains  a  multitude  of  minute  particles  in  suspension.  An 
example  of  this  is  the  sky-blue  liquid  obtained  by  adding  to  water 
a  very  small  proportion  of  milk  or  an  alcoholic  solution  of  mastic, 
or  by  mixing  a  few  drops  of  dilute  nitrate  of  silver  with  a  quantity 
of  water  in  which  a  little  table  salt  has  been  dissolved.^    These 

1  In  this  case  chloride  of  silver  is  formed,  which  is  insoluble  in  water,  but 
remains  suspended  in  the  form  of  extremely  minute  solid-^Ef&^e^r=-^K^^same  is 
true  of  the  mastic.  jf^^*    ^^  Twr 

'UNIVERSITY 


330  Light 

liquids  appear  blue  by  reflected  light ;  but  are  yellow  or  orange 
when  viewed  by  transmitted  light.  This  is  due  to  the  fact  that 
the  suspended  particles  reflect  a  considerable  part  of  the  violet 
and  blue  light,  but  reflect  less  and  less  of  the  other  colors  toward 
the  red  end  of  the  spectrum.  Thus  violet  and  blue  predominate 
in  the  reflected  light,  and  red,  orange,  and  yellow  in  the  trans- 
mitted light. 

The  blue  color  of  the  sky  is  similarly  explained,  "  the  air  being 
rendered  visible  against  the  dark  background  of  black  space  by 
sunlight  reflected  from  its  fine  suspended  dust  or  water  particles ; 
while  the  light  transmitted  is  always  more  or  less  yellowish,  and, 
in  the  afternoon  and  evening,  when  sunlight  comes  to  us  through  a 
greater  thickness  of  the  more  dusty  layers,  verges  toward  orange 
or  even  red." —  r3anieirs  PrincipUs  of  Physics. 

410.  Mixture  of  Colors.  — The  unaided  eye  is  wholly  incapable 
of  distinguishing  between  composite  and  elementary  colored  light. 
The  light  reflected  from  a  piece  of  yellow  paper  or  transmitted 
through  yellow  glass  may  produce  exactly  the  same  color  sensation 
as  the  yellow  of  the  spectrum,  although  it  contains  all  the  colors 
of  the  spectrum  from  the  red  to  the  green  inclusive.  Moreover, 
the  sensation  of  yellow  may  be  caused  by  light  that  contains  no 
elementary  yellow  at  all ;  in  fact,  the  color  sensation  caused  by  any 
elementary  color  except  violet  and  red  may  also  be  caused  by  vari- 
ous combinations  of  two  or  more  of  the  other  elementary  colors 
in  certain  proportions.^  In  studying  mixtures  of  colored  lights,  the 
selected  colors  of  the  spectrum  are  focused  by  a  lens,  or  reflected 
by  mirrors  to  the  same  spot  upon  a  screen,  or  in  some  other 
way  are  caused  to  enter  the  eye  in  a  united  beam.  Among  the 
results  established  by  such  experiments  are  the  following :  — 

I.  The  mixture  of  any  two  elementary  colors  named  in  alter- 
nate spaces  in  Fig.  248  is  like  the  intermediate  one.   For  example, 

1  If  the  ear  were  like  the  eye  in  this  respect,  it  would  be  incapable  of  distinguish- 
ing the  constituents  of  a  complex  sound.  The  notes  sounded  simultaneously  by 
an  orchestra  would  produce  ^e  sensation  of  a  single  note  of  average  pitch,  and 
harmony  and  discord  would  alike  be  unknown. 


Dispersion  and  Color 


331 


U99Jl'J 

Fig.  248. 


Z»^^ 


the  mixture  of  red  and  yellow  appears  to  the  eye  to  be  identical 

with  elementary  orange.     The  mixture  of  red  and  violet  light  is 

purple.       There    is   no   elementary 

,  .  •  ,  r  ,        r  Purple 

purple,  as  is  evident  from  the  fact 

that  this  color  is  not  found  in  the 

spectrum. 

2.  Complementary  Colors.  —  The 
mixture  of  any  pair  of  opposite  col- 
ors in  the  figure  appears  white  when 
the  colors  are  combined  in  the  right 
proportions.  **  Ordinary  white  light 
consists  of  all  the  colors  of  the  spec- 
trum combined  ;  but  any  one  of  the 
elementary  colors,  from  the  extreme 
red  to  a  certain  point  in  yellowish  green,  can  be  combined  with 
another  elementary  color  on  the  other  side  of  green  in  such 
proportions  as  to  yield  a  perfect  imitation  of  ordinary  white.  The 
prism  would  instantly  reveal  the  differences,  but  to  the  naked  eye 
all  these  whites  are  completely  undistinguishable  from  one  another." 
(Deschanel.) 

Any  two  colors  which  yield  white  when  mixed  are  called  com- 
plemeniary  colors, 

3.  Primary  Colors.  —  Any  color  of  the  spectrum  can  be  pro- 
duced (so  far  as  the  color  sensation  is  concerned)  by  mixing 
elementary  red,  green,  and  violet  lights  in  proper  proportions. 
Red,  green,  and  violet  are  therefore  called  the  primary  colors. 

411.  Newton's  Disks.  —  One  of  the  most  convenient  methods  of 
mixing  colors  is- by  means  of  colored  disks,  each 
of  which  is  slit  along  a  radius,  thus  permitting 
any  desired  amount  of  overlapping  when  two 
or  more  of  the  disks  are  placed  together  on 
an  axis  through  their  common  center  (Fig. 
249).  When  the  disks  are  rapidly  rotated 
about  the  axis,  only  one  color  is  seen,  and  this 
Fig.  249.  covers  the  entire  circular  area.     This  results 


332  Light 

from  the  fact  that  the  sensation  of  sight  continues  for  a  fraction 
of  a  second  after  the  light  ceases  to  enter  the  eye  or  ceases  to  fall 
upon  the  same  part  of  the  retina.  The  rapid  rotation  causes  the 
different  colors  to  come  from  all  parts  of  the  surface  in  such  rapid 
succession  that  each  of  them  produces  a  continuous  impression ; 
and  the  effect  is  the  same  as  if  they  came  simultaneously  from  all 
parts  of  the  surface.  , 

The  colors  reflected  by  the  disks  are,  of  course,  composite; 
but  experiments  have  shown  that  a  composite  color  produces  the 
same  effect  in  a  mixture  as  the  elementary  color  that  looks  like 
it.  Hence  the  results  described  in  the  preceding  article  can  be 
obtained  with  the  disks.  In  most  cases,  however,  the  total  amount 
of  reflected  light  is  so  small  that  the  resultant  color  is  very  defi- 
cient in  brightness.  Complementary  colors,  for  example,  generally 
yield  a  dark  gray  instead  of  white  {Exp.). 

412.  After-images.  —  If  one  looks  for  half  a  minute  or  more  at 
a  brightly  illuminated  piece  of  colored  paper  on  a  black  back- 
ground, then  at  a  white  surface,  an  image  of  the  colored  paper 
will  appear  upon  the  surface  in  the  complementary  color.  Thus  if 
the  paper  is  green,  the  image  will  appear  purple  {Exp.). 

The  explanation  is  that  the  part  of  the  retina  upon  which  the 
light  from  the  colored  paper  falls  becomes  fatigued  for  that  color, 
and  is  less  sensitive  to  it  than  to  the  other  colors  of  white  light ; 
hence  these  other  colors  produce  the  stronger  impression  when 
white  light  falls  upon  that  part  of  the  retina,  and  they  together 
give  the  complementary  color.  The  image  thus  produced  is  called 
a  negative  after-image. 

When  the  object  looked  at  is  very  bright,  the  after-image  is 
u'S\i2\\y positive,  that  is,  of  the  same  color  as  the  object;  and  this 
is  frequently  followed  by  a  negative  image.  A  positive  after- 
image niay  be  regarded  as  an  extreme  instance  of  the  persistance 
of  impressions.  After-images  of  bright  objects  can  often  be  seen 
with  the  eyes  tightly  closed.  Newton  is  said  to  have  suffered  for 
many  years  from  an  after-image  of  the  sun,  caused  by  incautiously 
looking  at  it  through  a  telescope. 


Dispersion  and  Color         --        333 

413.  Colors  of  Mixed  Pigments.  —  We  have  seen  that  a  mixture 
of  blue  and  yellow  lights  (either  composite  or  elementary)  is  white 
(Arts.  410  and  411)  ;  but  that  when  pieces  of  blue  and  yellow 
glass  are  placed  together,  they  appear  green  (Art.  408,  end).  The 
results  are  not  inconsistent,  for  they  are  obtained  by  wholly  differ- 
ent methods  of  combination.  In  the  first  case  the  sensation  of 
white  is  due  to  the  simultaneous  action  of  the  two  colors  upon  the 
retina ;  the  colors  and  the  sensations  that  they  produce  are  added. 
If  the  colors  are  complex,  as  is  the  case  with  Newton's  disks,  they 
together  probably  include  all  the  elementary  colors.  In  the  sec- 
ond case,  the  result  is  obtained  by  subtraction^  elementary  green 
being  the  only  color  that  passes  through  both  pieces  of  glass  in 
appreciable  quantity.  The  other  colors  are  absorbed,  —  some  by 
the  blue  glass,  the  remainder  by  the  yellow. 

The  mixture  of  blue  and  yellow  paints  or  powders  is  green 
{Exp.).  This  is  evidently  a  case  of  subtraction,  like  that  of  the 
blue  and  yellow  glass.  The  blue  paint  (or  powder)  absorbs  the 
red,  orange,  and  yellow  of  the  incident  light,  and  the  yellow  paint 
absorbs  the  violet,  indigo,  and  blue.  Thus  green  is  the  only  color 
not  strongly  absorbed  by  one  or  the  other,  and  the  mixture  is 
green.  It  is  evident  that  the  results  obtained  by  mixing  pigments 
and  by  mixing  lights  of  the  same  color  are,  in  general,  very  differ- 
ent. The  light  reflected  by  mixed  pigments  consists  of  the  colors 
which  are  not  absorbed  by  either  constituent.  If  the  lights  reflected 
by  two  pigments  have  no  elementary  color  in  common,  a  mixture 
of  the  two  will  be  black  or  a  dark  gray.  This  is  the  case  with 
vermilion  (a  bright  red)  and  ultramarine  (a  deep  blue)   {Exp.). 

414.  Chromatic  Aberration.  —  The  pupil  has  very  probably 
observed  a  fringe  of  color  bordering  objects  when  viewed  through 
a  convex  lens,  and  especially  when  viewed  through  a  pair  of 
lenses,  used  either  as  a  telescope  or  a 
compound  microscope.  The  coloring 
is  due  to  the  unequal  refraction  of 
the  elementary  colors  by  the  lenses.  '  ^^°* 

When  white   light  from  any  point,   O  (Fig.  250),  falls  upon  a 


334  Light 


lens,  the  greater  refraction  of  the  violet  light  brings  it  to  a  nearer 
focus,  Vy  than  that  of  the  red  light,  r.  The  foci  of  the  other  colors 
lie  between  these.  When  a  screen  is  placed  at  the  focus  of  the 
red  light,  the  image  is  surrounded  by  a  border  of  violet  and  blue 
light ;  at  the  focus  of  the  blue  light  it  is  surrounded  by  red  {Exp.), 
This  unequal  focusing  of  the  different  colors  is  called  chromatic 
aberration. 

Since  dispersion  increases  with  the  deviation,  chromatic  aberra- 
tion is  much  greater  for  light  passing  through  a  lens  near  its  edge 
than  for  light  passing  through  its  central  portion,  and  is  greater  the 
less  the  focal  length  of  the  lens  {Exp.).  Thus  the  opaque 
diaphragm  with  a  small  circular  opening,  which  is  placed  in  front 
of  the  lens  in  cameras  and  other  optical  instruments,  serves  the 
double  purpose  of  diminishing  both  the  spherical  and  the  chro- 
matic aberration.  But  the  diaphragm  only  partially  overcomes 
the  defect,  and  besides  has  the  disadvantage  of  cutting  off  the 
greater  part  of  the  light.  During  the  seventeenth  century  the 
remedy  employed  in  the  construction  of  telescopes  was  the  use  of 
object  glasses  of  great  focal  length — in  some  cases  exceeding 
lOo  feet.  The  object  glass  in  such  cases  was  mounted  at  the  top 
of  a  high  pole,  and  the  eyepiece  was  on  a  separate  stand  below. 
415.  Achromatic  Lenses.  —  About  150  years  after  the  telescope 
and  the  microscope  were  invented,  it  was  discovered  that  chro- 
matic aberration  could  be  almost  perfectly  corrected  by  combining 
a  convex  lens  of  crown  glass  with  a  concave  lens  of  flint  glass  (Fig. 
251).  This  depends  upon  the  fact  that,  while  the  refracting  power 
of  flint  glass  is  only  slightly  greater  than  that  of 
crown  glass,  its  dispersive  power  is  nearly  twice 
as  great,  the  spectrum  formed  by  a  prism  of  flint 
Fig.  251.  glass  being  nearly  twice  as  long  as  that  formed 

by  a  prism  of  crown  glass  having  an  equal  refracting  angle  {Exp.). 
Hence  a  double  lens,  consisting  of  a  convex  lens  of  crown  glass 
and  a  concave  lens  of  flint  glass  of  nearly  twice  the  focal  length, 
produces  convergence  of  the  light  without  dispersion ;  for  the  dis- 
persion due  to  the  concave  lens  is  equal  and  opposite  to  that 


Dispersion  and  Color 


335 


of  the  convex  lens,  and  it  neutralizes  only  half  of  the  converg- 
ence caused  by  the  latter  (Fig.  252).  The  objectives  of  tele- 
scopes, opera  glasses,  and  microscopes 
(except  the  cheapest)  are  achromatic. 

416.  The  Rainbow.  —  Rainbows  are  due 
to  the  dispersion  of  sunHght  by  raindrops, 
and  by  the  drops  of  water  in  the  spray  of 
fountains,  waterfalls,  etc.     Sometimes  one 


fc 


c 
Fig.  25a. 


bow  is  seen,  sometimes  two,  each  consist- 
ing of  the  colors  of  the  solar  spectrum. 
They  are  always  arcs  of  circles,  and,  when 
two  are  seen,  they  are  concentric.  The  ~ 
inner  or  lower  one  is  always  much  the  _ 
brighter  and  is  called  the  primary'  bow. 
In  it  the  red  is  on  the  outside,  the  violet 
on  the  inside.  In  the  outer,  or  secondary  bowy  the  colors  occur  in 
the  reverse  order  (Fig.  253).  The  rainbow  is  always  seen  in  the 
direction  opposite  to  the  sun,  —  the  sun,  the  observer,  and  the 
center  of  curvature  of  the  bow  being  in  the  same  straight  line,  EO. 
This  line  is  called  the  axis  of  the  bow. 

The  formation  of  the  rainbow  can 
be  experimentally  illustrated  in  a  dark- 
ened room  by  means  of  a  globe  (a 
round-bottomed  flask)  filled  with  water. 
It  will  be  found  that  the  results  obtained 
,  ..,  when  a  slender  beam  of  sunlight  is 
^    «o-«HM  ^^^"^  I ',  \   caused  to  fall  upon  the  globe  depend 

upon  the  angle  at  which  the  beam  meets 
its  surface.  At  a  certain  angle  a  curved 
spectrum  is  formed  and  may  be  caught  upon  a  screen  ;  at  another 
angle  the  spectrum  reappears  with  the  colors  in  the  reverse  order. 
At  angles  other  than  these  the  light  is  too  widely  scattered  on 
leaving  the  globe  to  produce  visible  effects.  If  a  beam  large 
enough  to  cover  the  globe  is  used,  one,  and  possibly  both,  of  the 
spectra  will  appear  as  complete  circles  {Exp.) . 


336 


Light 


42 


Fig.  254. 


The  dispersion  caused  by  a  single  drop  of  water  is  like  that 
obtained  with  the  globe  in  the  above  experiment,  except  that 
the  total  amount  of  reflected  light  is  correspondingly  less.  Each 
drop  forms  two  complete  spectra ;  but  the  eye  receives  only  a 
slender  ray  of  one  color  from  any  one  drop,  and  the  bow  that  is 
seen  is  made  up  of  light  from  a  multitude  of  drops. 

417.  The  Primary  Bow.  —  Figure  254  represents  a  ray  of  light 
entering  a  drop  at  the  angle  required  for  the  primary  bow.     The 

unequal  refraction  of  the  colors  at  A 
'X  causes  dispersion,  forming  a  spec- 
trum, ^  y,  at  the  back  of  the  drop. 
Here  the  greater  part  of  the  light  is 
refracted  out  (not  shown  in  the 
figure)  ;  the  remainder  is  reflected 
to  /i'y,  where  the  greater  part  of 
it  is  refracted  out  with  further  dis- 
persion. The  red  and  the  violet 
rays  make  angles  of  about  42**  and  40°,  respectively,  with  the  inci- 
dent ray  SA.  The  reason  for  the  order  of  the  colors  in  the 
primary  bow  will  be  evident  from  a  comparison  of  this  figure  with 
the  preceding  one.  The  eye  receives  the  same  color  fi-om  all 
drops  at  the  same  angular  distance  from  the  axis  of  the  bow ; 
hence  the  bow  is  circular. 

At  sunrise  or  sunset  the  rainbow,  if  complete,  appears  as  a  semi- 
circle, its  axis,  £0(Fig.  253),  being 
horizontal.  Since  the  center  of  the 
bow  is  always  at  the  same  angle  be- 
low the  horizon  that  the  sun  is  above 
it,  the  higher  the  sun  is,  the  shorter 
will  be  the  arc  of  the  bow.  When 
the  sun  is  more  than  42°  above  the 
horizon,  the  primary  bow  is  wholly 
below  it,  and  is  therefore  invisible. 

418.  The  Secondary  Bow.  — The       ^'  ^'''  '^* 
secondary  bow  is  formed  by  light  that  has  undergone  two  reflec- 


Dispersion  and  Color  337 

tions,  as  shown  in  Fig.  255.  On  leaving  the  drop,  the  red  ray 
makes  an  angle  of  about  51°,  and  the  violet  ray  an  angle  of  54°, 
with  the  incident  ray.  A  comparison  of  Figs.  255  and  253  will 
show  why  the  secondary  bow.  is  above  the  primary,  and  why  the 
order  of  the  colors  is  reversed.  The  faintness  of  the  secondary 
bow  is  due  to  the  additional  loss  of  light  at  the  second  reflection. 

419.  Color  by  Interference.  —  When  two  pieces  of  plate  glass  are 
pressed  firmly  together  in  the  fingers  or  in  a  clamp,  curved  bands 
of  spectrum  colors  appear,  surrounding  the  point  where  contact  is 
closest.  The  colors  are  brightest  when  the  plates  are  looked  at 
from  the  more  strongly  illuminated  side  {Exp.). 

In  Fig.  256,  MM'  and  NN'  represent  sections  of  the  glass 
plates,  the  distance  between  them  being  greatly  exaggerated. 
Light  incident  along  the  path  AB  is  par- 
tially reflected  at  C  from  the  lower  sur- 
face of  the  upper  plate,  and  also  at  E  from 
the  upper  surface  of  the  lower  plate.  Some 
of  the  light  reflected  at  E  is  transmitted 
through  the  upper  plate  parallel  to  and 
nearly  coincident  with  the  light  reflected 
from  C.     But,  in  twice  crossing  the  space  ^^^'  ^^^' 

between  the  plates^  the  waves  reflected  at  E  fall  behind  those 
reflected  at  C;  and  the  waves  of  some  one  of  the  elementary 
colors  in  these  two  sets  of  reflected  waves  meet  in  opposite  phase, 
causing  interference  as  in  the  case  of  sound  waves  (Art.  2C)8),  and 
that  color  is  weakened  or  destroyed.  The  reflected  light  is  com- 
plementary to  the  light  cut  out  by  interference,  and  this  differs  at 
different  places,  depending  upon  the  distance  between  the  plates. 

Patches  and  bands  of  rainbow  colors  are  similarly  produced  by 
the  interference  of  light  reflected  from  the  two  surfaces  of  a  very 
thin  film  of  any  transparent  substance,  as  a  soap  film  or  a  film  of 
oil  floating  on  water.  In  the  above  experiment  the  layer  of  air 
between  the  glass  plates  acts  as  a  thin  film.  Bodies  whose  colors 
are  due  to  the  interference  of  the  light  reflected  from  them  are 
called  iridescent. 


338  Light 

The  iridescence  of  some  bodies  is  caused  by  interference  of  the 
light  reflected  from  minute  parallel  grooves  and  ridges  (striations) 
covering  their  surfaces,  as  in  mother-of-pearl  and  the  plumage  of 
many  birds.  The  colors  of  an  iridescent  surface  change  with  the 
angle  of  incidence  of  the  light,  producing  the  beautiful  effect 
known  as  a  **  play  of  colors." 

The  interference  of  light  is  the  strongest  evidence  in  support  of 
the  wave  theory. 

PROBLEMS 

1.  \Miat  is  the  function  of  the  lens  in  producing  a  pure  spectrum? 

2.  Why  is  it  not  possible  to  correct  the  chromatic  aberration  of  a  lens  by 
any  change  in  the  form  of  its  surfaces? 

3.  (<i)  State  all  the  conditions  necessary  for  a  rainbow,  (d)  Do  two  ob- 
■ervers  see  exactly  the  same  rainbow  ? 

4.  Prove  that  the  reflection  within  a  raindrop  takes  place  at  less  than  the 
critical  angle,  and  is  therefore  not  total. 


CHAPTER   XI 

MAGNETISM 

I.  Properties  of  Magnets 

420.  Natural  and  Artificial  Magnets.  —  A  magnet  is  a  body 
which  has  the  property  of  attacting  iron,  and  the  term  magnetism 
is  appHed  to  the  unknown  cause  of  this  attraction. 

Natural  magnets,  or  lodestonesy  were  known  to  the  ancients. 
They  are  heavy,  black  stones  consisting  of  a  certain  iron  ore 
called  magnetic  oxide  of  iron.  The  word  magnet  is  derived  from 
Magnesia  in  Asia  Minor,  where  these  magnetic  stones  were  first 
found ;  they  were  called  lodestones  (leading-stones)  from  their 
property  of  pointing  in  a  definite  direction  when  suspended. 

When  a  piece  of  highly  tempered  steel  is  rubbed  with  a  magnet 
or  in  any  other  way  subjected  to  strong  magnetic  action,  it 
acquires  permanent  magnetic  properties,  and  becomes  a  manu- 
factured or  artificial  magnet.  Magnets  are  made  of  various  shapes, 
each  adapted  to  particular  uses. 

Laboratory  Exercise  63, 

421.  Distribution  of  Attracting  Power  ;  Polarity.  —  The  differ- 
ent parts  of  a  magnet  have  very  unequal  power  of  attracting  iron. 
This  is  readily  shown  by  placing 

the   magnet   in   a   box   of  iron 

filings  or  small  tacks.     On  lifting 

the  magnet,  it  will  be  found  that 

Fig.  257. 
the  quantity  of  filings  or  tacks 

clinging  to  the  magnet  is  greatest  at  and  near  the  ends,  and 

diminishes  rapidly  toward  the  center,  where  there  are  none  (Fig. 

257).     The  regions  near  the  ends,  where  the  force  of  attraction 

is  greatest,  are  called  \\iQ  poles  {Exp). 

339 


340  Magnetispi 

Every  magnet  that  is  regularly  or  uniformly  magnetized  has 
two  poles,  one  at  each  end,  whatever  its  shape  may  be.  By 
irregular  magnetization  additional  poles  may  be  developed  between 
the  ends ;  but  such  cases  need  not  be  considered. 

Any  straight  magnet,  when  suspended  or  supported  so  that  it  is 
free  to  turn  in  a  horizontal  plane,  always  comes  to  rest  with  the 
same  end  pointing  nearly  north.  This  end  of  the  magnet  is  called 
the  north  pole  (i>.  north-seeking  pole)  and  the  other  end  the  south 
poh  {Exp,). 

422.  The  Bfagnetic  Needle.  —  A  slender  magnet  suspended  at 

its  center  by  an  untwisted  fiber 
or  balanced  on  a  pivot  (Fig. 
258)  is  called  a  magnetic  needle. 
After  any  displacement,  a  mag- 
netic needle  always  returns  to 

*°*  ^^  *  one  definite  direction,  which 

is  the  direction  of  the  magnetic  meridian  at  that  place. 

423.  The  Mutual  Action  of  Ifagnets.  —  Any  force  exerted  be- 
tween two  magnets  is  most  readily  detected  when  one  or  both  are 
free  to  turn  about  an  axis  under  the  action  of  the  force.  Thus 
when  either  pole  of  a  bar  magnet  is  brought  up  to  the  like  pole  of 
a  magnetic  needle,  the  latter  turns  away  from  the  magnet,  show- 
ing repulsion  ;  but  the  unlike  pole  turns  toward  the  magnet,  show- 
ing attraction  (Exp.).  When  the  experiment  is  performed  with 
two  needles,  both  will  turn,  showing  that  the  attractions  and  repul- 
sions are  mutual.  The  law  of  equal  action  and  reaction  (third 
law  of  motion)  holds  for  magnetic  forces  as  for  all  others. 

The  mutual  action  of  magnets  is  expressed  by  the  law :  Like 
poles  repel  and  unlike  poles  attract  each  other. 

424.  The  Effect  of  Distance  on  Bfagnetic  Action.  —  The  force 
exerted  by  a  pole  of  a  magnet  increases  as  the  distance  from  the 
pole  decreases.  This  is  shown  by  the  fact  that  the  nearer  an  end 
of  a  bar  magnet  is  brought  to  a  magnetic  needle,  the  more  rapidly 
will  the  unlike  pole  of  the  needle  swing  round  and  vibrate  before 
it     ITie  exact  relation  between  the  force  and  the  distance  varies 


Properties  of  Magnets  341 

considerably  under  different  conditions ;  but  the  force  is  approxi- 
mately inversely  proportional  to  the  square  of  the  distance  except 
for  very  short  distances. 

This  accounts  for  the  fact  that,  when  the  nearer  poles  of  two 
magnets  are  unlike,  the  magnets  tend  to  come  together ;  for  the 
attraction  exerted  upon  the  nearer  pole  of  either  exceeds  the 
repulsjon  exerted  upon  its  farther  pole,  giving  a  resultant  force 
which  acts  toward  the  other  magnet.  Similarly,  when  the  nearer 
poles  are  like,  the  resultant  force  upon  each  magnet  tends  to 
move  it  from  the  other. 

425.  Magnetic  and  Nonmagnetic  Substances.  —  Substances 
that  are  attracted  by  a  magnet  are  called  magnetic  substances ; 
those  that  are  not  attracted  are  called  nonmagnetic.  The  only 
substances  that  are  sufficiently  magnetic  to  be  attracted  by  mag- 
nets of  ordinary  strength  are  iron,  steel  (a  form  of  iron),  some 
compounds  of  iron,  including  magnetic  iron  ore,  nickel,  and  cobalt. 
All  other  substances  are  practically  nonmagnetic ;  although,  under 
the  action  of  a  powerful  electro-magnet  (Art.  457),  all  or  nearly 
all  exhibit  either  weak  attraction  or  repulsion.  Nickel  and  cobalt 
are  less  magnetic  than  iron.  Iron  in  its  different  forms,  such  as 
cast  iron,  wrought  iron,  and  steel,  is  the  only  substance  whose 
magnetic  properties  are  of  any  importance. 

These  properties  are  usefully  applied  in  the  telegraph,  the  tele- 
phone, the  dynamo,  the  motor,  and  many  other  electrical  machines 
and  instruments. 

426.  Magmetic  Action  through  Bodies.  —  The  action  of  a  mag- 
net upon  any  magnetic  body  is  not  affected  by  placing  a  non- 
magnetic body  between  them ;  but  is  affected  in  a  marked  degree 
by  interposing  another  magnetic  body.  For  example,  a  magnet 
attracts  or  repels  a  magnetic  needle  through  a  board,  a  book,  or  a 
plate  of  glass  just  as  if  nothing  intervened  ;  but,  when  a  sheet  of 
iron  is  placed  between  them,  the  needle  is  only  slightly  affected 
by  the  presence  of  the  magnet,  if  at  all.  The  sheet  of  iron,  espe- 
cially if  large,  serves  as  a  screen  to  cut  off  magnetic  action  from  the 
side  opposite  the  magnet.     This  is  explained  in  the  next  article. 


342  Magnetism 

n.   Kagnetization 

Laboratory  Exfrcise  64. 

427.  Bfagnetic  Induction  ;  Penneability.  —  When  either  pole  of 
a  magnet  is  held  against  or  very  near  an  end  of  a  soft  iron  rod, 
the  other  end  of  the  rod  attracts  iron  filings  in  considerable  quan- 
tity, and  attracts  or  repels  the  poles  of  a  magnetic  needle  as  the 
nearer  pole  of  the  magnet  would  do.  Thus  with  the  north  pole 
against  the  rod  (Fig.  259),  the  farther  end  of  the  rod  repels  the 
north  pole  of  the  needle.  The  rod,  in  fact,  senies  as  a  carrier 
for  the  magnetic  action  of  the  magnet^  and  is  itself  a  magnet  while 
doing  so,  having  poles  as  shown  in  the  figure.  Only  magnetic 
substances  can  thus  modify  and  extend  the  action  of  a  magnet, 
and  the  property  thus  exhibited  is  called  magnetic  permeability. 
We  can  now  understand  how  a  sheet  of  iron  serves  as  a  magnetic 
screen.  By  itself  becoming  magnetized,  it  turns  aside  the  mag- 
netic action  to  its  edges,  which  afe  capable  of  exerting  attractions 
and  repulsions  (Exp.). 

The  iron  rod  in  the  above  experiment  is  said  to  be  magnetized 
by  induction.     Magnetic   induction   always   takes   place  when   a 

^_ —  '  magnet  is  brought  near  a  magnetic 

•*?  -v    body.    The   body  is  then  attracted 

Fig.  259.  because  its  nearer  pole  is  unlike  the 

nearer  pole  of  the  magnet  (Fig.  259).     Thus  induction  always 
precedes  attraction  and  is  the  cause  of  it. 

428.  Temporary  and  Permanent  Kagnets.  —  When  soft  or 
wrought  iron,  hard  iron,  untempered  steel,  and  tempered  steel 
are  subjected  to  equal  inductive  action,  as  by  bringing  them  suc- 
cessively in  contact  with  the  same  magnet,  a  simple  test  with  iron 
filings  will  show  that  the  soft  iron  becomes  most  strongly  mag- 
netized and  the  tempered  steel  the  least.  On  the  other  hand,  the 
soft  iron  loses  its  magnetization,  completely  or  nearly  so,  as  soon 
as  it  is  removed  from  the  influence  of  the  magnet,  while  the  tem- 
pered steel  retains  all  or  nearly  all  of  the  magnetization  induced 
in  it.    Thus  by  magnetic  induction  a  piece  of  soft  iron  may  be 


Magnetization 


343 


made  a  temporary  magnet  and  a  piecfe  of  tempered  steel  a  perma- 
7ient  one.  Hard  iron  and  untempered  steel  retain  a  considerable 
part  of  induced  magnetization  ;  they  are  subpermafient. 

All  manufactured  magnets  are  pieces  of  highly  tempered  steel 
that  have  been  magnetized  by  induction.  They  may  be  demag- 
netized or  even  magnetized  with  opposite  polarity  at  any  time  by 
sufficiently  strong  inductive  action  in  the  opposite  direction. 

429.  Experimental  Evidence  on  the  Nature  of  Magnetization. — 
There  is  much  experimental  evidence  indicating  that  the  magneti- 
zation of  a  body  is  definitely  related  to  its  molecular  condition. 
The  following  are  some  of  the  facts  that  point  most  strongly  to 
this  conclusion :  — 

Properties  of  a  Broken  Magnet.  —  When  a  magnet  is  broken 
into  any  number  of  parts,  each  piece  is  a  complete  magnet  having 
the  same  polarity 
and  approximately 
the  same  strength 
as  the  original  mag- 
net (Fig.  260).  This 
can  be  readily  shown  by  breaking  a  magnetized  sewing  or  knitting 
needle  {Exp,). 

Such  experiments  show  that  every  part  of  a  magnet  is  mag- 
netized ;  in  fact,  the  neutral  portion  in  the  middle  is  generally 
more  strongly  magnetized  than  the  ends.  The  absence  of  attract- 
ive power  in  the  middle  may  be  accounted  for  by  regarding  the 

two  halves  of  a  magnet 

^  ^-^ as     complete    magnets 

with  their  unlike  poles 
joined  at  the  center. 
These  equal  unlike  poles, 
situated  at  the  same 
point,  exactly  neutralize  each  other's  action  upon  surrounding 
bodies.  A  magnet  may  be  regarded  as  composed  of  an  in- 
definite number  of  little  magnets,  with  all  their  like  poles  pointing 
in  the  same  direction  (Fig.  261). 


!„i;i;i;li;:ii:;^J 

Fig.  260. 

^1 

M— 

» 

If- 

—9 

n 

s 

7f- 

■  8 

n 

« 

n 

8 

n 

9 

n 

8 

n 

8 

n 

8 

n 

s 

n 

8 

n       8 

n 

8 

n 

8 

n 

8 

n 

S 

n 

8 

n 

8 

n 

8 

n 

8 

n 

8 

n 

8 

n 

3 

n 

H 

n 

S 

n 

-2 

n      8 

2- 

-2 

n 

8 

n 

8 

7l 

-J» 

N 


Fig.  261. 


344  Magnetism 

Effect  of  Heat  on  Magnetization,  —  The  strength  of  a  magnet  is 
always  diminished  by  heating  it.  This  effect  is  only  temporary  for 
moderate  degrees  of  heat ;  but  a  bright  red  heat  causes  permanent 
demagnetization  {Ex/>,), 

This  effect  of  heat  suggests  that  magnetization  depends  upon  or 
consists  in  a  certain  molecular  condition  which  is  destroyed  by 
the  violent  motion  of  the  molecules  at  a  high  temperature. 

Effect  of  Mechanical  Disturbance.  —  A  magnet  is  weakened  by 
hitting  it  a  number  of  sharp  blows,  and  a  magnetized  knitting 
needle  by  clamping  it  in  a  vise  and  causing  it  to  vibrate  vigorously 
(Exp.).  A  magnetized  piece  of  iron  wire  a  foot  or  so  in  length 
is  almost  completely  demagnetized  by  twisting  it  once  or  twice 
each  way.  (A  small  portion  at  each  end  may  be  bent  at  right 
angles  to  the  length  for  convenience  in  twisting.)  {Exp.) 

In  such  cases  a  loss  of  magnetization  results  from  the  disturbance 
of  the  molecular  condition  of  the  magnet  by  mechanical  forces. 
On  the  other  hand,  such  disturbances  assist  magnetic  forces  in 
producing  magnetization.  Thus  a  number  of  blows  upon  a  piece 
of  iron  when  it  is  near  a  magnet  will  cause  it  to  become  more 
strongly  magnetized. 

430.  Theory  of  Magnetization.  —  These  facts  and  others  of  a 
similar  nature  have  led  to  the  theory  that  every  molecule  of  a 
magnetic  substance  is  a  permanent  magnet ;  and  that  in  an  unmag- 
netized  body  the  poles  of  these  molecular  magnets  point  indis- 
criminately in  all  directions  (Fig.  262),  while  in  a  magnetized 

body  the  greater 
number  of  the  mole- 


r^^^^M^^M^i^ 


!?,«  ^.^  ...^     ,  cules  lie  with   their 

Fia  a63.  (IG.  263. 

like  poles  pointing 
in  the  same  direction  (Fig.  263).  According  to  this  theory, 
the  act  of  magnetizing  consists  in  turning  the  molecules  more 
or  less  completely  into  one  particular  direction.  If  all  the  mole- 
cules were  turned  in  the  same  direction,  the  limit  of  possible 
magnetization  would  be  reached.  Soft  iron  is  more  readily 
magnetized  than  steel,  because  its  molecules  are  more  easily  turned 


The  Magnetic  Field     .  345 

about,  and  it  loses  its  magnetization  more  readily  for  the  same 
reason.  This  theory  may  be  illustrated  by  means  of  a  test  tube 
nearly  full  of  steel  filings,  each  particle  of  which  plays  the  part  of  a 
molecule  on  a  greatly  magnified  scale.  When  the  mass  of  filings 
is  magnetized,  it  exhibits  polarity  and  acts  as  a  magnet  until 
shaken  up  (Lab.  Ex.).  What  constitutes  the  magnetism  of  the 
molecule  is  not  known. 

III.   The  Magnetic  Field 

431.  Magnetic  Field  and  Lines  of  Force.  —  A  magnetic  field  is 
any  space  within  which  magnetic  forces  act.  The  intensity  of 
these  forces  is  often  referred  to  as  the  intensity  of  the  tnagnetic 
field.  The  field  of  a  magnet  is  most  intense  near  the  poles,  and 
decreases  rapidly  with  increasing  distance.  It  really  extends 
indefinitely  in  every  direction ;  but  at  a  comparatively  short  dis- 
tance it  becomes  too  weak  to  produce  sensible  effects. 

When  a  magnetic  needle  is  placed  within  the  field  of  a  magnet, 
as  at  O  (Fig.  264),  its  north  pole  is  attracted  by  the  south  pole 
of  the  magnet  and  repelled  by  the  north  pole.     OB  represents 
the  attraction  and  OA  the  re- 
pulsion upon  the  north  pole  of 
the  needle  when  at  0\  and  OR 
their  resultant  (by  the  parallelo- 
gram  of  forces).      Hence    the 
needle  behaves  as  if  its   north 
pole  were  acted  upon  by  a  single    '  pj^^    ,  ' 

force  in  the  direction  of  OR, 
The  component  and  resultant  forces  upon  the  south  pole  of  the 
needle,  when  at  O^  are  respectively  equal  in  magnitude  and 
opposite  in  direction  to  those  upon  the  north  pole ;  hence  a 
short  needle  placed  with  its  center  at  O  would  come  to  rest 
with  its  north  pole  pointing  in  the  direction  OR. 

If  the  needle  is  moved  constantly  in  the  direction  in  which  its 
north  pole  points,  it  will  trace  the  curved  path  OCS,  the  direction 


346  Magnetism 

of  which  at  any  point  is  the  direction  of  the  resultant  magnetic  force 
at  that  point.  Such  a  line  in  a  magnetic  field  is  called  a  line  of 
force.  Since  the  resultant  forces  upon  a  north  and  a  south  pole  are 
in  opposite  directions  along  a  line  of  force,  we  avoid  ambiguity  by 
defining  the  direction  of  a  line  of  force  as  the  direction  of  the 
force  acting  upon  a  north  pole.  In  accordance  with  this  defini- 
tion, the  lines  of  force  in  the  field  of  a  magnet  are  said  to  extend 
fix)m  its  north  to  its  south  pole  (not  from  the  south  to  the  north 
pole).  In  diagrams  the  direction  of  a  line  of  force  is  often  indi- 
cated by  an  arrowhead  placed  on  the  line. 
Laboratory  Exercise  dj, 

432.   Lines  of  Force  in  Certain  Magnetic  Fields.  —  The  direction 
of  the  lines  of  force  in  different  parts  of  a  magnetic  field  can  be 

determined  by  merely  observing  the 
direction  in  which  the  north  pole  of  a 
magnetic  needle  points  when  moved 
about  in  the  field  ;  but  fine  iron  filings 
may  be  made  to  serve  the  same  pur- 
pose for  the  whole  of  a  plane  section 
of  a  field  at  one  time.  The  filings  are 
sprinkled  from  a  pepper  box  or  other 
sifter  upon  a  sheet  of  cardboard  or 
stiff  paper  placed  over  the  magnet. 
*  A   light    tapping  on    the    cardboard 

assists  the  magnetic  forces  in  bringing  the  filings  into  definite  lines 
coinciding  with  lines  of  force. 

The  lines  thus  obtained  about            .  \  j  //v'""^"^Vv  •  A'V 
a   bar  magnet  are  shown   in  •  \\\\'.l /v;^rr:::-^> '.'■'»!/'/// ^ 

«=»  — >^r- — > — 15>»» — ->       f 

Fig.  265.     Beyond  a  distance  ,y;,'  j ; \\.'^:~~".-s^y/i]\ v^\ 

.of  a  few  inches   the  field  is  'V i  \ 'v^srrirl'Cr' V  i  \  X'* 

too  weak  to  direct  the  filings ;  ^ 

hence    the    information    that  ^'^*  ^^' 

they  afford  is  incomplete.  All  the  lines  of  force  extending  from 
the  north  pole  are  really  continuous  with  lines  of  force  coming  to 
the  south  pole  (when  no  other  magnet  is  in  the  vicinity). 


N 


TTT 
/// 


The  Magnetic  Field  347 

When  the  north  pole  of  one  magnet  is  placed  near  the  south  pole 
of  another,   the    filings    are 
arranged  in  lines  as  shown  in 
Fig.  266.     The  lines  of  force 
extend   across    between   the  i\^ 

unlike  poles  of  the  two  mag-  \  \^  ^ .  \  >    /  '   / ,  /  i 

nets.    But  when  the  like  poles  \  ^.  \  \  "'i  \  /  >^  /  /  /  ' 

of  two  magnets  are   turned  i   i  \  '.  .  .'      .  .  •  . 

toward  each  other  (Fig.  267),  * 

no  lines  are  found  to  extend  from  one  to  the  other ;  on  the  con- 
trary, the  lines  in  one  field  turn  away  from  those  of  the  other. 

433.  Theory  of  Magnetic  Action.  —  Magnetic  action  takes  place 
at  a  distance  apparently  without  the  aid  of  any  medium  by  means 
of  which  the  force  is  exerted  upon  the  distant  body.  In  this 
respect  it  is  like  gravitation ;  and,  like  gravitation,  it  takes  place 
in  a  vacuum.  But  as  action  at  a  distance  without  an  intervening 
medium  for  its  transmission  is  not  considered  possible  (Art.  129), 
it  is  assumed  that  a  medium  exists  by  means  of  which  magnetic 
forces  are  exerted,  and  this  medium  is  thought  to  be  no  other  than 
the  ether  that  pervades  all  space. 

According  to  this  view,  the  ether  surrounding  a  magnet  is  under 
certain  stresses  (Art.  206)  due  in  some  way  to  the  presence  of  the 
magnet;  and  the  laws  of  magnetic  action  indicate  that  these 
stresses  consist  of  a  tension  along  the  lines  of  force  and  a  pressure 
across  them.  Thus  we  have  a  mental  picture  of  an  elastic  sub- 
stance which  is  in  a  state  of  tension  between  unlike  poles,  tending 
to  draw  them  together  by  contraction  along  the  lines  of  force,  and 
in  a  state  of  compression  between  like  poles,  tending  to  push  them 
apart  by  expansion  at  right  angles  to  the  lines  of  force. 

PROBLEMS 

I,  (a)  When  a  pole  of  a  strong  magnet  is  brought  toward  the  like  pole 
of  a  magnetic  needle,  repulsion  may  be  followed  by  attraction  as  the  magnet 
is  brought  closer.  Explain.  {U)  The  same  may  happen  when  an  end  of  a 
weakly  magnetized  piece  of  iron  is  brought  toward  a  needle.     Explain. 


348  Magnetism 

2.  Why  should  decision  as  to  the  polarity  of  a  magnetized  body  be  based 
on  repulsion  of  the  magnetic  needle  rather  than  on  attraction  ? 

3.  In  what  different  ways  may  an  unmagnetized  magnetic  substance  be 
dbtinguished  from  a  magnet  ? 

4.  Explain  the  effect  of  the  soft  iron  bar,  called  the  armature  or  keeper, 
which  is  placed  across  the  ends  of  a  horseshoe  magnet,  when  not  in  use,  to 
preserve  the  strength  of  the  magnet. 

5.  In  what  respects  does  magnetic  action  resemble  gravitation  ?  In  what 
respects  docs  it  differ  ? 

IV.  Terrestrial  Magnetism 

434.  The  Earth's  Magnetic  Field.  —  Everywhere  upon  the 
earth's  surface  the  magnetic  needle,  when  removed  from  all  mag- 
netic substances,  always  comes  to  rest  in  a  definite  direction, 
clearly  indicating  that  it  is  controlled  by  a  magnetic  field.  This  is 
the  magnetic  field  of  the  earth.  It  varies  largely  in  intensity  and 
in  the  direction  of  its  lines  of  force  over  different  portions  of  the 
earth's  surface ;  but  the  changes  are  so  gradual  that  the  lines  of 
force  are  sensibly  straight  and  parallel,  and  the  intensity  constant 
over  areas  many  miles  in  extent.  In  general,  the  lines  of  force 
extend  in  a  direction  several  degrees  either  to  the  east  or  west  of 
north,  and  are  more  or  less  inclined  to  the  horizontal. 

The  magnetic  field  of  the  earth  indicates  that  the  earth  is  an 
irregularly  magnetized  body.  The  cause  of  its  magnetization  is  not 
known. 

435.  Action  of  the  Earth's  Field  on  a  Compass  Needle.  —  How- 
4Aaa*aaaa*4**a»*4   ever  much  or  little  the  lines  offeree  of 

I    the  earth's  field   may  be  inclined,  it  is 

I   only   the   horizontal   component   of  the 

I    magnetic   force   that   exerts  a   directive 

j    action  upon  magnets  that  are  free  to  turn 

only  in  a  horizontal  plane.     Hence  the 

Fig.  268.  directive  action   upon  compass  needles 

is  the  same  as  it  would  be  if  the  lines  of  force  were  horizontal, 

as  represented  in  Fig.  268. 

The  horizontal  forces  exerted  upon  the  two  poles  of  a  compass 


Terrestrial  Magnetism 


349 


needle  by  the  earth's  magnetic  field  are  equal  and  opposite,  since 
the  field  is  of  uniform  intensity  and  the  lines  of  force  are  straight. 
Hence  when  the  needle  is  not  parallel  to  these  forces,  they  act 
as  a  couple  to  swing  the  needle  into  Hne  with  them  (A  and  B, 
Fig.  268) ;  and  the  needle  is  brought  to  rest  in  this  position  by 
friction  after  a  number  of  vibrations. 

436.  Magnetic  Meridians  and  Declination.  —  A  line  extending 
over  the  earth  and  having  at  every  point  the  direction  of  the  com- 
pass  needle    is   called  a  magnetic  meridian.     In   Fig.    269  the 


somewhat  irregular  heavy  lines  represent  magnetic  meridians. 
The  magnetic  meridian  at  any  place  is  sometimes  called  the  mag- 
netic north-and-south  line ;  and  the  angle  that  it  makes  with 
the  true  north-and-south  line  is  called  the  magnetic  declina- 
tion or  simply  the  decHnation  at  the  place  considered.  The  curved 
Hues  in  Fig.  270  are  so  drawn  that  each  passes  through  all  points 
having  the  same  declination.  Such  lines  are  called  isogenic  lines, 
or  lines  of  equal  declination.  The  arrows  in  the  figure  show  the 
direction  of  the  declination,  whether  east  or  west. 

Magnetic  declination  is  subject  to  a  number  of  variations,  only 
one  of  which  exceeds  a  small  fraction  of  a  degree.  This  variation 
consists  in  a  continuous  change  in  declination  in  one  direction  for 
about  two  hundred  years,  followed  by  a  like  change  in  the  oppo- 


350 


Magnetism 


Fig,  aTa 


site  direction.     Recorded  observations  show  that  such   changes 

have  amounted  to  over  35°  in  certain  localities. 
437.   Inclination  or  Dip;   Magnetic   Poles  of  the  Earth. —  A 

magnetic  needle  mounted  on  a  horizontal  axis  through  its  center 
of  gravity  is  called  a  dipping  needle  (Fig. 
271).  If  a  dipping  needle  were  unmag- 
netized,  it  would  be  in  neutral  equilibrium 
in  any  position  in  th6  vertical  plane  in  which 
it  is  free  to  turn  ;  hence  its  position  in  the 
vertical  plane  is  controlled  only  by  magnetic 
forces,  as  is  the  position  of  the  compass 
needle  in  a  horizontal  plane.  Consequently, 
I- 10.  271.  when  the  axis  of  a  dipping  needle  is  placed 

at  right  angles  to  the  magnetic  meridian,  the  needle  comes  to  rest 

in  the  direction  of  the  lines  of  force  of  the  earth's  field. 

The  angle  between  the  direction  of  the  dipping  needle  and  the 

horizontal  is  called   the  inclination  or  dip.     The   irregular   lines 

extending  across  Fig.  272  are  lines  of  equal  dip.    The  line  of  no 


Terrestrial   Magnetism 


351 


dip  is  called  the  magnetic  equator.  North  of  the  magnetic  equator 
the  north  pole  of  the  needle  is  depressed,  and  south  of  it  the 
south  pole. 

Arctic  explorers  have  found  a  place  where  the  dip  is  90°  ;  this 
is  the  north  magnetic  pole  of  the  earth  (so  called  from*  its  geo- 


FlG.  272. 

graphical  position,  not  from  its  polarity).  It  is  nearly  1400  miles 
from  the  geographical  north  pole,  and  is  shown  in  Fig.  269  as 
the  point  in  the  northern  hemisphere  to  which  the  magnetic 
meridians  converge.  The  south  magnetic  pole  has  never  been 
reached.  Strictly  speaking,  the  magnetic  poles  of  the  earth  are 
far  below  the  surface. 

438.  Intensity  of  the  Earth's  Field;  Inductive  Action. — The 
earth's  magnetic  field  is  much  too  weak  to  arrange  iron  filings  in 
lines  or  to  appreciably  modify  magnetic  action  within  a  few  cen- 
timeters of  a  strong  magnet ;  but  it  is  everywhere  sufficiently 
intense  to  control  a  magnetic  needle  and  to  cause  considerable 
magnetization  in  iron  and  steel.  This  inductive  action  is  best 
shown  by  means  of  a  long  rod  of  soft  iron  (Norway  iron).     When 


352  Magnetism 

such  a  rod  is  held  so  as  to  point  north  and  south  or,  better,  in  the 
direction  of  the  dipping  needle,  it  will  be  found  to  be  magnetized 
with  its  north  pole  pointing  north.  When  the  rod  is  reversed,  its 
polarity  is  also  instantly  reversed,  if  the  iron  is  very  soft ;  other- 
wise it  may  be  necessary  to  strike  the  rod  on  the  end  while  it  is 
held  in  position  {Exp.). 

Any  mass  of  iron  or  steel  that  remains  in  one  position  for  a 
long  time  becomes  magnetized.  This  is  especially  true  of  the 
rails  of  a  track  when  extending  in  a  northerly  and  southerly 
direction.  Natural  magnets  are  very  probably  due  to  the  earth's 
induction. 

439.  Importance  of  the  Earth^s  Magnetism.  —  The  importance 
of  the  earth's  magnetism  is  due  to  its  directive  action  on  the  com- 
pass needle.  This  is  utilized  on  land  in  determining  directions 
in  surveying,  and  on  sea  in  directing  the  course  of  vessels.  In 
the  use  of  the  compass  for  either  6f  these  purposes  the  declination 
at  the  place  must  be  known.  At  sea  this  is  given  by  the  declina- 
tion map  or  chart  (Fig.  270)  from  the  known  latitude  and  longi- 
tude of  the  vessel. 

"  Neither  the  inventor  of  the  compass  nor  the  exact  time  of  its 
invention  is  known.  Guyot  de  Provins,  a  French  poet  of  the 
twelfth  century,  first  mentions  the  use  of  the  magnet  in  navigation, 
though  it  is  probable  that  the  Chinese  long  before  this  had 
used  it.  The  ancient  navigators,  who  were  unacquainted  with  the 
compass,  had  only  the  sun  or  pole  star  as  a  guide,  and  were 
accordingly  compelled  to  keep  constantly  in  sight  of  land  for  fear 
of  steering  in  a  wrong  direction  when  the  sky  was  clouded."  — 
Ganot's  Physics. 

The  earth's  magnetic  field  also  plays  an  essential  part  in  the 
use  of  certain  instnmients  (galvanometers)  in  electrical  measure- 
ments, as  will  be  seen  later. 


CHAPTER  XII 
ELECTRICITY 

440.  "  Electricity  and  magnetism  are  not  in  themselves  forms 
of  energy ;  neither  are  they  forms  of  matter.  They  may  perhaps 
be  provisionally  defined  as  properties  or  conditions  of  matter ;  but 
whether  this  matter  be  the  ordinary  matter,  or  whether  it  be,  on 
the  other  hand,  that  all-pervading  ether  by  which  ordinary  matter 
is  everywhere  surrounded  and  permeated,  is  a  question  which  has 
been  under  discussion,  and  which  is  now  held  to  be  settled  in 
favor  of  the  latter  view."  —  Daniell's  Principles  of  Physics, 

I.   The  Voltaic  CeU 

441.  Action  of  Dilute  Sulphuric  Acid  on  Zinc  and  Copper. — 
When  a  strip  of  zinc  is  placed  in  dilute  sulphuric  acid  (one  part 
by  volume  of  acid  to  fifteen  or  twenty  of  water),  it  is  acted  upon 
by  the  acid  and  is  gradually  dissolved  or  eaten  away.  This 
action  is  accompanied  by  the  rapid  formation  of  small  bubbles, 
which  adhere  to  the  zinc  until  detached  by  the  buoyant  force  of 
the  liquid.  The  escape  of  these  bubbles  at  the  surface  gives  the 
liquid  the  appearance  of  boiling  (Lab.  Ex.). 

»  These  effects  are  the  result  of  chemical  action.  Sulphuric  acid 
is  a  compound  substance,  the  constituents  of  which  are  hydrogen, 
sulphur,  and  oxygen.  The  action  of  the  acid  upon  the  zinc  con- 
sists in  the  substitution  of  zinc  for  tlie  hydrogen  of  the  acid,  by 
which  a  compound  (zinc  sulphate)  consisting  of  zinc,  sulphur, 
and  oxygen,  is  formed  and  the  hydrogen  of  the  acid  set  free.  The 
bubbles  observed  in  the  experiment  are  bubbles  of  hydrogen  ;  the 
zinc  sulphate  remains  in  solution  in  the  liquid.  ,The  union  of 
the  zinc  and^he  acid  liberates  chemical  potential  energy,  which, 

353 


354 


Electricity 


under  the  conditions  of  the  experiment,  is  transformed  into  heat, 
—  much  as  the  energy  of  coal  is  transformed  into  heat  by  union 
with  oxygen  in  the  process  of  burning. 

When  copper  is  placed  in  dilute  sulphuric  acid,  no  bubbles  are 
formed  and  the  copper  does  not  waste  away  however  long  it  may 
remain  in  the  liquid.     There  is  no  appreciable  chemical  action. 

442.  The  Simple  Voltaic  Cell.  —  When  a  strip  of  zinc  and  a 
strip  of  copper  are  in  the  same  vessel  of  dilute  sulphuric  acid,  but 

are  not  in  contact,  the  appear- 
ance of  each  strip  is  the  same 
as  if  the  other  were  not  pres- 
ent—  bubbles,  of  hydrogen 
form  upon  the  zinc,  but  not 
upon  the  copper.  But  on 
connecting  the  strips  by  means 
of  a  wire  soldered  to  each 
(Fig.  273),  bubbles  form  upon 
the  copper  as  well  as  upon  the 
zinc.  There  is  still,  however, 
no  chemical  action  upon  the  copper,  as  is  evident  from  the  fact  that 
the  copper  does  not  waste  away  however  long  the  strips  may 
remain  connected. 

If  the  wire  connecting  the  strips  be  turned  so  as  to  extend 
north  and  south,  a  compass  needle  held  close  to  it,  either  above 
or  below,  will  be  deflected,  indicating  the  existence  of  a  magnetic 
field  about  the  wire.  This  magnetic  field  indicates  that  something 
is  happening  within  and  about  the  wire  as  the  result  of  its  connec- 
tion with  the  zinc  and  copper  strips  in  the  acid  ;  and  other  effects 
can  be  produced  which  prove  conclusively  that  there  is  a  transfer- 
ence of  energy  along  the  wire  (either  through  the  wire  or  through  the 
ether  surrounding  it)  from  one  of  the  metal  strips  to  the  other. 
For  example,  an  electric  bell  may  be  rung,  a  telegraph  sounder 
operated,  or  a  piece  of  platinum  wire  heated  by  making  proper 
connections  with  the  wires  attached  to  the  strips. 

These  effects  and  others  are  attributed  to  what  is  known  as  an 


Fig.  273. 


The  Voltaic  Cell 


355 


electric  current  flowing  through  the  wire  from  the  copper  strip  to 
the  zinc,  and  through  the  liquid  from  the  zinc  to  the  copper, 
making  thus  a  complete  circuit.  We  speak  of  electricity  as  if  it 
were  a  fluid,  and  think  of  it  as  flowing  through  wires  and  other 
conductors  in  much  the  same  way  as  water  flows  through  pipes ; 
but  it  is  not  known  that  anything  is  actually  transferred  round  an 
electric  circuit  except  energy  (of  the  kind  known  as  electrical 
energy),  nor  is  it  certain  in  which  direction  the  transference  of  this 
energy  takes  place.  It  is  assumed,  however,  that  the  direction  of 
the  current  is  as  stated  above.  There  is  always  a  magnetic  field 
about  a  conductor  carrying  a  current  of  electricity ;  this  is  some- 
times called  an  electro-magtietic  field y  because  it  is  due  to  electricity 
instead  of  magnetism.  The  magnetic  fields  of  electric  currents 
serve  as  the  readiest  means  of  detecting  and  measuring  them,  and 
also  give  rise  to  most  of  the  industrial  applications  of  electricity. 

The  vessel  of  acid,  together  with  the  zinc  and  copper  strips,  is 
called  a  voltaic  cell^  in  honor  of  Alessandro  Volta^  an  Italian  physi- 
cist, who  devised  this  method  of  producing  an  electric  current  in 
1800.^  Two  or  more  cells  connected  together  constitute  a  voltaic 
or  electric  battery^.  In  popular  usage,  a  single  cell  is  commonly 
called  a  battery.  The  copper  strip  is  called  \.\\&  positive  plate,  pole, 
or  electrode  of  the  cell,  and  the  zinc  strip  the  negative.  In  diagrams 
the  signs  -f  and  —  are  often  used  to  indicate  the  positive  and 
negative  plates  respectively. 

Laboratory  Exercise  66. 

443.  Cause  of  the  Current ;  Potential  and  Electro-motive  Force. 
—  In  a  voltaic  cell  such  as  we  have  described,  the  energy  of  the 
current  is  derived  from  the  chemical  action  of  sulphuric  acid  on 
zinc.  How  this  chemical  action  causes  an  electric  current  is  a 
theoretical  question  that  can  hardly  be  discussed  with  profit  in  an 
elementary  course.  It  should  be  remembered,  however,  that  while 
the  cell  is  in  action,  one  of  the  constituents  of  sulphuric  acid 
(hydrogen)  goes  to 'the  copper  plate,  where  it  accumulates  as 

1  The  Xerm  galvanic  is  sometimes  used  instead  of  voltaic;  it  is  derived  from  the 
name  of  Galvani,  an  Italian  physician,  who  discovered  current  electricity  in  1786. 


356  Electricity 

bubbles  of  visible  size,  and  that  the  other  part  of  the  acid,  consist- 
ing of  sulphur  and  oxygen  in  chemical  union,  goes  to  the  zinc 
plate,  with  which  it  immediately  unites,  forming  zinc  sulphate. 
More  or  less  hydrogen  is  also  liberated  at  the  surface  of  the  zinc, 
depending  upon  its  condition;  but  this  is  not  essential  to  the 
action  of  the  cell  and  results  only  in  wasted  energy  (Art.  447). 

It  can  be  shown  experimentally  that  the  zinc  and  the  copper 
plates  differ  in  respect  to  a  condition  known  as  electrical  potential ; 
and  the  copper  is  said  to  be  at  a  higher  potential  than  the  zinc* 
The  difference  between  the  potentials  of  the  plates  is  the  direct 
cause  of  the  current  when  the  plates  are  connected  by  a  wire.^  A 
difference  of  potential  can  be  maintained  between  two  parts  of  a 
circuit  by  other  means  than  chemical  action  (by  a  dynamo,  for 
example) ;  but,  however  this  may  be  brought  about,  the  result  is 
always  an  electric  current  from  the  point  at  higher  to  the  point  at 
lower  potential  when  the  points  are  connected  by  any  conductor. 
This  behavior  of  electricity  may  be  compared  to  the  conduction 
of  heat  between  two  points  from  higher  to  lower  temperature,  or 
to  the  flow  of  water  in  a  pipe  from  a  higher  to  a  lower  level. 

The  agency  that  moves  electflcity  from  any  point  to  a  point  at 
a  lower  potential  is  called  electro-motive  force  (often  denoted  by 
E.  M.  F.).  "  Just  as  in  water  pipes  a  difference  of  level  produces 
a  pressure y  and  the  pressure  produces  2iflow  as  soon  as  the  tap  is 
turned  on,  so  difference  of  potential  produces  electro-motive  force ^ 
and  electro-motive  force  sets  up  a  current  as  soon  as  the  circuit  is 
completed  for  the  electricity  to  flow  through."  Electro-motive  force 
and  difference  of  potential  are  commonly  used  as  equivalent  expres- 
sions ;  we  shall  have  no  occasion  to  distinguish  between  them.  It 
should  be  noted  that  electro- motive  force  is  not  a  force  at  all,  in 
the  proper  sense  of  the  word,  since  it  does  not  act  upon  matter, 
but  upon  electricity. 

•  1  When  the  zinc  and  copper  plates  are  connected  by  a  wire,  there  is  a  continu- 
ous fall  of  potential  round  tlie  circuit  in  the  direction  of  the  current  (through  the 
liquid  from  the  zinc  to  the  copper  plate  as  well  as  through  the  wire),  except  in 
passing  from  the  zinc  to  the  liquid  and  from  the  liquid  to  the  copper,  at  each  of 
which  places  there  is  an  abrupt  rise  of  potentiaL  The  potential  of  the  zinc  is  the 
lowest  in  the  entire  circuit 


The  Voltaic  Cell 


157 


444.  Conductors  and  Nonconductors ;  Resistance.  —  All  sub- 
stances offer  greater  or  less  opposition  to  the  passage  of  electricity 
through  them,  and  the  property  thus  exhibited  is  called  electrical 
resistance.  The  resistance  of  a  good  conductor  is  small,  that  of  a 
poor  conductor  large.  Substances  through  which  electricity  is 
nearly  or  wholly  unable  to  pass  are  called  nonconductors  or 
insulators. 

The  metals  are  good  conductors  compared  with  other  substances, 
but  differ  largely  among  themselves,  copper  being  much  the  best 
with  the  exception  of  silver.  Carbon  and  dilute  acids  are  the 
next  best  conductors,  though  not  nearly  so  good  as  metals.  The 
resistance  of  the  liquid  in  a  voltaic  cell  is  often  much  greater  than 
that  of  the  remainder  of  the  circuit.  Silk,  india  rubber,  vulcanite, 
and  glass  are  some  of  the  best  insulators. 

445.  The  Electric  Circuit.  —  A  current  of  electricity  requires  a 
complete  circuit,  i.e.  a  path  that  is  continuous  from  any  point 
back  to  that  point  again  without  retracing  any  portion  of  it.  The 
circuit  may  be  made  up  of  any  number  of  substances,  and  these 
m'ay  be  of  any  size  or  shape.  It  is  only  necessary  that  all  parts 
of  the  circuit  be  of  materials  capable  of  conducting  electricity  and 
that  they  be  placed  in  close  contact.  The  circuit  is  said  to  be 
closed  when  it  is  complete,  open  or  broken  when  there  is  a  gap  at 
any  point. 

The  wire  of  an  electric  circuit  is  either  insulated  by  a  non- 
conducting cover  or  supported  upon  insulators  (usually  of  glass) 
to  prevent  the  escape  of  the  electricity  from  the  path  intended  for 
it.  For  currents  of  low  potential,  such  as  are  used  in  ringing 
electric  bells,  a  cotton  covering  affords  sufficiently  good  insula- 
tion;  for  higher  potentials  the  covering  is  of  silk  or  rubber.  In 
connecting  wires  for  a  circuit,  the  insulation  must  be  removed  for 
a  short  distance  at  the  ends,  and  the  bare  wires  fastened  together. 

446.  Materials  used  in  Voltaic  Cells.  —  An  electric  current  may 
be  obtained  with  various  dilute  acids  and  solutions  of  different 
silts,  and  the  plates  may  be  made  of  any  two  metals  that  are 
unequally  acted  upon  by  the  liquid.     The  greater  the  difference 


358  Electricity 

in  the  chemical  action  upon  the  plates,  the  greater  will  be  their 
difference  of  potential,  and  hence  also  the  greater  will  be  the 
current.  The  best  results  are  therefore  obtained  when  the  nega- 
tive plate  is  made  of  zinc  and  the  positive  plate  of  copper  or 
carbon ;  for  zinc  is  most  readily  acted  upon  by  the  acids  and 
solutions  used  in  different  cells,  and  copper  and  carbon  are  not 
acted  uf)on  at  all. 

447.  Local  Action  upon  the  Negative  Plate.  —  We  have  seen 
that  some  hydrogen  is  liberated  at  the  zinc  plate  when  a  simple 
voltaic  cell  is  in  action  (Art.  443,  first  paragraph).  This  is  due 
to  the  presence  of  small  particles  of  iron,  carbon,  or  other  impuri- 
ties in  commercial  zinc.  Any  such  particle  on  the  surface  of  the 
zinc  and  in  contact  with  the  liquid  acts  as  a  positive  pole  and 
forms  a  minute  voltaic  cell  with  the  adjacent  zinc  and  liquid. 
This  causes  a  local  or  parasitic  current  at  the  spot  and  a  continual 
wasting  of  the  zinc,  whether  the  circuit  is  open  or  closed. 

When  the  zinc  used  is  chemically  pure,  this  local  action ^  as  it 
is  called,  does  not  occur.  The  zinc  is  consumed  only  when  the 
circuit  is  closed,  and  hydrogen  bubbles  apj)ear  only  upon  the 
copper.  The  same  result  is  obtained  with  a  plate  of  commercial 
zinc  when  amalgamated^  i.e.  when  covered  with  a  coating  of 
mercury.  The  mercury  dissolves  a  portion  of  the  zinc,  forming 
a  pasty  amalgam  which  covers  the  surface  and  keeps  the  acid 
from  contact  with  the  impurities.  As  pure  zinc  is  expensive,  it  is 
more  economical  to  use  plates  of  commercial  zinc  and  keep  them 
amalgamated.  Amalgamation  is  necessary,  however,  only  in  the 
case  of  cells  in  which  acids  are  used,  as  in  the  chromic  acid  cell 
(Art.  449). 

448.  Polarization  of  the  Positive  Plate.  —  The  current  supplied 
by  one  or  more  cells  of  the  type  already  described  rapidly  dimin- 
ishes from  the  moment  the  circuit  is  closed.  This  may  be  shown 
by  measuring  the  current  with  a  galvanometer  (a  process  to  be 
considered  later),  or  by  arranging  a  battery  of  one  or  more  cells 
that  is  just  sufficient  to  ring  an  electric  bell,  operate  a  telegraph 
sounder,  or  run  a  small  motor  when  the  circuit  is  first  closed. 


The  Voltaic  Cell  359 

The  current  will  very  quickly  become  too  weak  to  produce  these 
effects.  The  power  of  the  battery  may  be  restored  by  removing 
the  positive  plates  and  wiping  them  thoroughly  or  by  letting  the 
battery  remain  idle  for  some  minutes  {Exp.). 

The  weakening  of  the  current  is  due  to  the  accumulation  of 
hydrogen  upon  the  positive  plate.  A  cell  that  is  in  this  condition 
is  said  to  be  polarized.  After  the  hydrogen  has  been  wiped  off  or 
has  had  time  to  escape,  the  current  is  as  strong  as  at  first. 
Polarization  causes  a  decrease  in  the  current  for  two  reasons.  In 
the  first  place,  hydrogen  is  a  nonconductor,  and  hence  increases 
the  resistance  of  the  cell  by  cutting  off  the  current  from  the  por- 
tion of  the  surface  that  it  covers.  In  the  second  place,  the  hydro- 
gen tends  to  reunite  with  the  other  constituents  of  the  acid,  just  as 
the  zinc  does,  though  less  strongly ;  and  hence  it  sets  up  an  oppos- 
ing electro-motive  force,  which  tends  to  send  a  current  in  the 
opposite  direction.  Polarization  is  avoided  or  diminished  in  vari- 
ous ways  in  different  forms  of  cells,  some  of  the  most  common  of 
which  are  described  in  the  following  articles.^ 

449.  Chromic  Acid  or  Bichromate  Cell.  —  The  zinc  plate  of  this 
cell  (Fig.  274)  is  attached  to  a  rod,  by  means  of 
which  it  can  be  raised  from  the  liquid  when  the 
cell  is  not  in  use.  The  positive  pole  consists  of 
two  plates  of  carbon  —  one  on  each  side  of  the 
zinc  —  which  are  connected  to  the  same  binding 
screw  at  the  top.  The  liquid  is  dilute  sulphuric 
acid,  containing  in  solution  chromic  acid  or  bi- 
chromate of  potassium  or  of  sodium,  which  acts 
as  a  depolarizer.  These  substances  contain  oxygen 
which  they  give  up  readily  to  hydrogen,  forming 
water.  The  accumulation  of  hydrogen  upon  the 
carbon  plates  is  thus  diminished,  but  not  entirely  ^^*  ^^ 

prevented.  Polarization  diminishes  the  current  by  one  third  or 
more  in  a  few  minutes. 

1  These  descriptions  may  be  most  profitably  studied  in  connection  with  the  use 
of  the  cells  in  the  laboratory  or  the  class  room.  Any  or  all  of  them  may  be  passed 
over  for  the  present. 


360  Electricity 

The  electro-motive  force  of  this  cell  is  large  compared  with  that 
of  most  cells.  Its  resistance  is  small,  for  the  current  has  only  a 
ver)'  short  path  in  the  liquid,  and  the  double  carbon  pole  reduces 
the  resistance  further  by  one  half.  As  a  result  of  the  high  E.  M.  F. 
and  low  resistance,  this  cell  is  capable  of  supplying  an  exception- 
ally strong  current,  and  on  this  account  is  much  used  in  experi- 
mental work.  The  zinc  should  be  kept  thoroughly  amalgamated  : 
it  must  be  raised  from  the  liquid  when  not  in  use^  for  it  is  attacked 
by  the  solution  even  when  the  circuit  is  open. 

450.  The  Leclanch^  Cell.  —  The  zinc  of  this  cell  is  usually  in  the 
form  of  a  rod  (Fig.  275)  ;  the  positive  plate  is  a  block  of  carbon. 

The  latter  is  inclosed  in  a  cylindrical  cup 
of  porous  earthenware,  and  is  surrounded 
by  small  fragments  of  carbon  and  man- 
ganese dioxide,  with  which  the  cup  is 
filled.  The  liquid  is  a  solution  of  am- 
monium chloride  (sal  ammoniac)  in  water. 
The  zinc  is  not  acted  upon  when  the  cir- 
cuit is  open,  and  does  not  require  amal- 
gamation. When  the  circuit  is  closed, 
chlorine  from  the  ammonium  chloride 
Fig.  275.  unites  with  the  zinc,  forming  zinc  chlo- 

ride and  liberating  the  other  constituents  of  the  ammonium  chloride 
(ammonia  and  hydrogen).  The  zinc  chloride  and  the  ammonia 
are  held  in  solution ;  the  hydrogen  passes  through  the  porous 
cup  and  unites  slowly  with  oxygen  from  the  manganese  dioxide, 
forming  water. 

This  cell  polarizes  rapidly,  and  is  suitable  only  for  uses  requiring 
brief  action  with  comparatively  long  intervals  of  rest,  during  which 
it  recovers  from  polarization.  It  is  much  used  for  ringing  electric 
bells,  and  has  the  merit  of  not  requiring  attention  for  months  at  a 
time. 

451.  The  Gravity  Cell.  —  The  positive  pole  of  this  cell  (Fig. 
276)  consists  of  a  number  of  strips  of  copper  fastened  together, 
and  is  placed  at  the  bottom  of  the  jar.     The  zinc  is  near  the  top, 


The  Voltaic  Cell 


361 


and  is  made  in  various  forms.  In  some  cases  it  is  supported  by  a 
rod,  as  in  the  figure  ;  in  others,  it  is  hung  from  the  edge  of  the  jar. 
The  lower  portion  of  the  hquid  is  a  strong 
solution  of  copper  sulphate  (blue  stone)  ; 
the  upper  portion  is  a  weak  solution  of 
zinc  sulphate/  which,  being  of  less  spe- 
cific gravity  than  the  lower  solution,  rests 
upon  it  without  mixing  except  by  the 
slow  process  of  diffusion. 

While  the  cell  is  in  use,  copper  from 
the  copper  sulphate  is  continually  de- 
posited upon  the  copper  plate.  By 
certain  actions  within  the  liquids  which 
need  not  be  considered,  the  other  con- 
stituent of  the  copper  sulphate  is  set  free  '  ^ 
at  the  zinc,  with  which  it  unites,  forming  zinc  sulphate.  The  solu- 
tion of  copper  sulphate  is  continually  renewed  from  a  supply  of 
crystals  of  copper  sulphate  placed  about  the  copper  plate.  When 
not  in  use,  this  cell  should  be  kept  on  a  closed  circuit  through  a 
considerable  resistance  (about  20  ohms). 
The  small  current  then  flowing  tends  to 
prevent  the  mixing  of  the  liquids  by  dif- 
fusion ;  otherwise  the  copper  sulphate, 
coming  in  contact  with  the  zinc  plate,  will 
deposit  copper  upon  it,  and  in  this  con- 
dition the  cell  will  furnish  little  or  no 
current.  The  zinc  is  not  amalgamated. 
Polarization  is  entirely  avoided  in  the 
gravity  cell,  and  the  current  that  it  sup- 
plies through  a  given  circuit  is  approxi- 
mately constant.  Its  E.  M.  F.  is  about 
half  that  of  the  chromic  acid  cell,  and  its 
resistance  is  several  times  as  great ;  the 
largest  current  obtainable  from  it  is  consequently  comparatively 
1  Water  containing  a  very  little  sulphuric  acid  may  be  used  in  setting  up  the  cell. 


Fig,  277. 


362  Electricity 

small.  It  is  especially  serviceable  in  experimental  work  requir- 
ing a  constant  E.  M.  F.,  and  is  much  used  for  purposes  requiring 
a  current  all  or  nearly  all  of  the  time,  as  in  telegraphy. 

452.  The  Daniell  Cell.  —  This  cell  is  essentially  the  same  as  the 
gravity  cell;  but  its  parts  are  differently  arranged  and  the  two 
solutions  are  kept  separate  by  a  partition  of  porous  earthenware 
in  the  form  of  a  cylindrical  cup  (Fig.  277).  The  porous  cup  con- 
tains the  negative  pole  (a  long  bar  of  zinc)  and  a  dilute  solution 
of  zinc  sulphate.  This  is  placed  in  a  glass  jar  containing  a  satu- 
rated solution  of  copper  sulphate.  The  positive  plate  is  a  sheet 
of  copper  surrounding  the  porous  cup. 

II.   The  Electro-magnetic  Field 

453.  Historical. — The  first  discovery  of  a  definite  relation 
between  electricity  and  magnetism  was  made  by  Oersted,  a  Danish 
physicist,  in  1819,  —  nineteen  years  after  Volta's  invention  of  the 
electric  battery.  He  found  that,  when  a  wire  carrying  a  current 
is  placed  in  a  horizontal  position  above  a  compass  needle,  the 
needle  is  deflected  from  the  magnetic  meridian.  When  the 
direction  of  the  current  is  reversed  or  the  wire  held  below 
the  needle,  the  deflection  is  in  the  opposite  direction. 

This  is  known  as  "  Oersted's  experiment."  It  marks  the  begin- 
ning of  the  science  of  eUctro-magnetismy  —  that  branch  of  electrical 
science  which  treats  of  the  relations  between  electricity  and  mag- 
netism and  which  is  usefully  applied  in  such  inventions  as  the 
electric  bell,  the  telegraph,  the  telephone,  the  dynamo,  and  the 
motor. 

Laboratory  Exercise  6/. 

454.  Magnetic  Field  due  to  a  Current  in  a  Straight  Conductor.  — 
Iron  filings  sprinkled  on  a  horizontal  piece  of  cardboard,  through 
which  a  vertical  conductor  passes,  show  circular  lines  of  force 
about  the  conductor  when  a  sufficiently  strong  current  is  flowing  ^ 

1  In  studying  the  magnetic  field  about  a  conductor  carrying  a  current,  a  strong 
current  must  be  used ;  otherwise  the  field  would  be  too  weak  to  bring  iron  filings 
into  line,  and  the  behavior  of  a  magnetic  needle  would  be  largely  modified  by  the 


The  Electro-magnetic  Field 


363 


Fig.  278. 


(Fig.  278).  Since  all  cross-sections  of  the  field  taken  at  right 
angles  to  the  conductor  are  alike,  it  will  be  seen  that  a  straight  con- 
ductor carrying  a  current  is  surrounded 
by  a  cylindrical  magnetic  field,  whose 
lines  of  force  are  circles  about  the  con- 
ductor as  a  center  and  lie  in  planes  at 
right  angles  to  it.  The  relation  between 
the  direction  of  the  lines  of  force  round 
the  wire  {i.e.  the  direction  in  which  the 
north  pole  of  a  needle  points)  and  the 
direction  of  the  current  is  imix)rtant  and 
may  easily  be  remembered  from  the  following  right-hand  rule: 
Grasp  the  wire  with  the  right  hand  so  that  the  extended  thumb 
points  in  the  direction  of  the  current ;  then  the  fingers  point  round 
the  wire  in  the  direction  of  the  lines  of  force. 

455.  Field  Due  to  a  Current  in  a  Circular  Coil.  —  When  a  con- 
ductor is  curved,  the  lines  of  force  are  crowded  together  on  the 
concave  side  and  spread  apart  on  the  convex  side,  the  planes  of 
the  lines  of  force  being  perpendicular  to  the  con- 
ductor at  every  point  (Fig.  279)  ;  hence  the  field 
is  relatively  strong  at  the  center  of  a  circular  coil, 
where  lines  of  force  round  all  parts  of  the  coil 
meet.  The  field  at  the  center  of  a  circular  coil  is 
of  special  interest  and  importance,  as  it  is  utilized 
in  the  simplest  form  o{ galvanometer  m  measuring 
electric  currents.  It  may  be  conveniently  studied 
by  sending  a  strong  current  through  all  the  turns 
(usually  fifteen)  of  a  galvanometer  coil.  Iron  filings  sprinkled 
on  a  piece  of  cardboard,  placed  within  the  coil,  will  show  that  the 
lines  of  force  at  the  center  of  the  coil  are  straight  and  are  perpen- 

magnetic  field  of  the  earth.  A  battery  of  three  or  four  chromic  acid  cells  connected , 
in  parallel  (Art.  481)  gives  a  current  sufficiently  strong  for  this  purpose;  but  even 
better  results  are  obtained  from  a  single  cell  by  sending  the  current  round  a  rec- 
tangle of  insulated  wire  cohsisting  of  six  or  eight  turns.  The  field  is  strengthened 
in  proportion  to  the  number  of  parallel  wires,  but  is  otherwise  the  same  as  if  there 
were  only  one. 


Fig.  279. 


364  Electricity 

dicular  to  the  plane  of  the  coil.  Their  direction  relative  to  the 
direction  of  the  current  is  given  by  the  following  right-hand  rult 
for  coils :  Close  the  right  hand  and  place  it  within  the  coil  with  the 
fingers  pointing  in  the  direction  of  the  current:  then  the  extended 
thumb  points  in  the  direction  of  the  lines  of  force  through  the  coil, 

456.  The  Helix  or  Solenoid.  ^  A  long,  cylindrical  coil  of  wire 
is  called  a  helix  or  solenoid.  The  successive  turns  of  the  coil  may 
be  at  Some  distance  apart  or  may  be  in  contact ;  in  the  latter  case 
insulated  wire  must  be  useci^  For  some  purposes  coils  are  wound 
upon  woo<len  spools  and  consist  of  several  layers  of  turns. 

The  magnetic  field  about  a  helix  through  which  a  current  is 
passing  is  similar  to  that  about  a  short,  flat  coil,  as  described  in 
the  preceding  article;  but  its  shape  is  somewhat  modified  (Fig. 

280).  Within  the  coil,  the  lines 
of  force  extend  from  end  to  end 
in  approximately  straight  lines. 
Without  the  coil,  the  field  is  like 
that  of  a  bar  magnet :  the  lines 
of  force  spread  out  from  one  end 
of  the  coil  and  return  to  the  other  ; 
at  one  end  the  south  pole  of  a  magnetic  needle  is  attracted,  at  the 
other  the  north  pole.  In  fact,  a  helix  through  which  a  current  is 
passing  may  be  regarded  as  a  magnet  and  as  having  a  north  and  a 
south  pole.  The  right-hand  rule  for  coils  may  be  conveniently 
stated  for  the  helix  as  follows :  Grasp  the  coil  so  that  the  fingers 
point  in  the  direction  of  the  current;  then  the  ex- 
tended thumb  points  in  the  direction  of  the  north  pole 
of  the  coil. 

When  a  helix  is  suspended  or  supported  horizon- 
tally in  such  a  way  that  it  is  free  to  turn  in  a  hori- 
zontal plane  while  a  current  is  flowing  through  it,  it 
'comes  to  rest  with  its  north  pole  pointing  north,  like 
a  compass  needle  {Exp). 

457.  The  Electro-magnet.  — When  a  soft  iron  rod  is 
placed  within  a  helix  through  which  a  current  is  floW'       fig.  «8i.~ 


Fig.  a8o. 


The   Electric   Bell  and  the  Telegraph      365 

ing  (Fig.  281),  it  becomes  strongly  magnetized,  its  polarity  being 
the  same  as  that  of  the  coil.  The  coil  and  iron  rod  together 
constitute  an  electro-magnet.  Electro-magnets  are  much  more 
powerful  than  steel  magnets  of  equal  size,  and  have  the  further 
advantage  that  their  magnetization  is  under  perfect  control,  for  they 
become  demagnetized  the  in- 
stant the  current  is  stopped. 
Electro-magnets  are  made 
of  different  shapes  for  differ- 
ent purposes,  various  modi- 
fications of  the  horseshoe 
form  (Fig.  282)  being  the 
most  generally  useful.     The 

current  flows  in  opposite  di- 

'  ^  Fig.  282. 

rections  round  the  two  coils 

of  a  horseshoe  magnet,  making  one  of  the  free  ends  of  the  iron 

core  north  and  the  other  south  according  to  the  rule  given  above. 

PROBLEMS 

1.  Apply  the  right-hand  rule  to  determine  the  direction  in  which  the  north 
pole  of  a  compass  needle  will  be  deflected  when  a  current  in  a  straight  con- 
ductor flows  from  north  to  south  directly  above  it;  directly  below  it  j  when 
the  current  flows  from  south  to  north  above  it ;  below  it. 

2.  What  would  be  the  angle  between  the  wire  and  the  direction  of  the 
needle  in  each  of  the  above  cases  if  the  field  of  the  earth  were  negligible  in 
comparison  with  that  of  the  current  ?  Why  would  the  angle  be  less  if  this 
were  not  the  case  ? 

3.  How  does  the  theory  of  magnetization  account  for  the  great  strength 
of  electro-magnets  ? 

4.  Does  the  current  in  a  helix  flow  clockwise  or  anti-clockwise  when  its 
north  pole  points  toward  the  observer  ?  when  the  south  pole  points  toward 
the  observer  ? 

III.   The  Electric  Bell  and  the  Telegraph 

458.  The  Electric  Bell.  — An  electric  bell  (Fig.  283)  is  rung 
by  the  action  of  an  electro-magnet.     Different  bells  are  somewhat 


366 


Electricity 


differently  constructed.     In  most  cases  the   metal  frame  of  the 
bell  forms  a  part  of  the  circuit ;  but  this  is  merely  a  matter  of 
»•  convenience   in  construction.     The  connec- 

tions and  insulations  must,  however,  be  such 
that  the  only  path  offered  the  current  through 
the  bell  is  by  way  of  the  coils  of  the  electro- 
magnet and  across  between  the  free  end  of  a 
spring,  a,  and  the  end  of  a  screw,  <*.  The 
spring  s,  which  carries  the  armature  of  the 
magnet  (a  soft  iron  bar),  draws  it  away  from 
the  magnet,  and,  at  the  same  time,  presses 
the  spring  a  against  the  screw.  (The  springs 
s  and  a  are  sometimes  in  one  piece,  as  shown 
in  the  figure.) 

When  the  circuit  is  closed  by  pressing  a 
push  button,  placed  at  some  convenient  point 
in  the  circuit,  the  electro-magnet  attracts  the 


Fig.  383. 


armature,  and  the  clapper  attached  to  it  strikes  the  bell.  At 
the  same  time  the  spring  is  pulled  away  from  the  screw  at  ^, 
breaking  the  circuit.  With  the  stopping  of  the  current,  the  mag- 
net becomes  demagnetized  and  ceases  to  attract  the  armature. 
The  spring  s  then  pulls  the  armature  back,  making  contact  at  c 
again.  This  process  is  repeated  as  long  as  the  button  is  pushed. 
Laboratory  Exercise  j6. 

459.  The  Electro-magnetic  Telegraph.  —  The  earliest  and  the 
simplest  system  of  transmitting  messages  by  electricity  is  the  elec- . 
tro-magnetic  telegraph.  The  credit  for  this  invention  belongs 
principally  to  Samuel  F.  B.  Morse  of  New  York,  who  devised  the 
first  practical  instruments.  The  first  telegraph  line  was  established 
by  Morse  in  1844  between  Washington  and  Baltimore.  The 
instruments  required  for  this  system  of  telegraphy  are  the  sounder^ 
the  key,  and  the  relay. 

460.  The  Sounder. — The  sounder  (Fig.  284)  has  an  electro- 
magnet, the  poles  of  which  point  upward.  An  armature  of  soft 
iron  is  fixed  across  a  lever  just  above  the  poles.     Whenever  the 


The  Electric  Bell  and  the  Telegraph      367 


current  passes  through  the  coils  of  the  electro-magnet,  the  arma- 
ture is  attracted  down,  carrying  the  lever  with  it,  and  a  screw 
near  the  end  of  the  lever 
makes  a  click  as  it  strikes 
against  a  stop.  As  soon  as 
the  current  ceases,  the  lever 
is  raised  by  a  spring  and 
strikes  against  the  end  of  a 
screw.  The  two  clicks  sound 
differently  and  are  easily  dis- 
tinguished. They  together 
constitute  a  "  dot  "  when  one  1-24. 

follows  immediately  after  the  other,  and  a  "dash  "  when  the  inter- 
val between  them  is  an  appreciable  fraction  of  a  second.  The 
letters  of  the  alphabet,  the  punctuation  marks,  and  the  numbers 
from  zero  to  nine  are  represented  by  different  numbers  of  dots, 
dashes,  and  various  combinations  of  the  two. 

461.  The  Key.  —  The  key  (Fig.  285)  is  a  device  by  which 
the  operator  makes  and  breaks  the  circuit  in  the  act  of  sending  a 
message.      It  is  fastened  to  a  table  by  two  screws,  the  one  shown 

at  the  left  in  the  figure  being 
insulated  from  the  metal 
base.  One  wire  of  the  line 
is  fastened  to  each  screw. 
There  is  a  small  platinum 
point  at  the  top  of  the  insu- 
lated screw,  and  another,  /*, 
just  above  it  ontl)e  under 
side  of  the  lever.  When 
the  lever  is  depressed,  these  points  come  in  contact,  closing  the 
circuit.  When  the  key  is  not  in  use,  the  circuit  is  kept  closed  by 
the  switch  S,  which  connects  the  base  of  the  instrument  with  the 
insulated  post,  as  shown  in  the  figure.  This  switch  is  moved  to 
the  right  while  a  message  is  being  sent,  leaving  the  circuit  open 
and  under  the  control  of  the  operator  by  means  of  the  lever. 


Fig.  285. 


368  Electricity 

462.  The  Relay.  — The  high  resistance  of  a  long  telegraph  line 
makes  the  current  too  weak  to  operate  a  sounder.  This  difficulty 
might  be  overcome  by  using  a  battery  of  a  very  great  number  of 

cells ;  but  it  is  more  conven- 
ient and  more  economical  to 

D l^^g       l^^^S^L.  i9>^^^-*     make  use  of  an  additional 

instrument  called  the  relay 
(Fig.  286).     The  relay  acts 
on  the  same  principle  as  the 
Fig.  286.  sounder,    but    the    electro- 

magnet is  horizontal  and  the 
armature  vertical.  The  lever  that  carries  the  armature  is  light 
and  delicately  balanced,  and  hence  is  easily  moved  to  and  fro  by 
much  smaller  forces  than  are  required  to  operate  the  lever  of  a 
sounder. 

The  coils  of  the  electro- magnet  are  connected  with  the  line  by 
means  of  the  binding  posts  A  and  B.  The  wires  of  a  local  circuit 
containing  the  sounder  and  a  battery  to  operate  it  are  connected 
to  the  posts  C  and  D.  One  of  these  posts  is  connected  by  a  wire 
under  the  base  of  the  instrument  with  a  metal  column,  the  upper 
end  of  which  forms  an  arch  above  the  lever,  and  the  other  post  is 
similarly  connected  with  the  lever.  The  local  circuit  is  closed  by 
contact  of  the  lever  with  the  platinum  tip  of  a  screw,  which  it 
strikes  when  drawn  over  by  the  electro-magnet,  and  is  broken 
when  the  magnet  ceases  to  act  and  the  lever  is  pulled  back 
by  a  spring  against  the  hard  rubber  tip  of  the  opposite  screw. 
Thus  when  the  line  circuit  is  closed  or  opened  by  means  of 
the  key,  the  local  circuit  is  at  the  same  instant  closed  or  opened 
by  the  action  of  the  relay.  Since  the  resistance  of  the  local 
circuit  is  small,  one  or  two  cells  are  sufficient  to  operate  the 
sounder.  The  sounds  made  by  the  lever  of  the  relay  are  nearly 
inaudible. 

463.  The  Telegraph  System.  —  A  diagram  of  a  complete  tele- 
graph system  connecting  two  cities  is  shown  in  Fig.  287.  A  key 
and  relay  are  included  in  the  main  line  at  each  station  (of  which 


Electrical  Measurements 


369 


there  may  be  any  number).  There  is  a  line  battery^  at  each  of 
the  terminal  stations,  each  consisting  of  many  cells  in  series 
(Art.  480),  and  the  two  are  so  connected  to  the  line  that  they 
exert  an  E.  M.  F.  in  the  same  direction  through  it.  Some  form  of 
cell  that  does  not  polarize  must  be  used,  as  the  gravity  cell.    The 


New  York 
Key  Sounder^ 


Philadelphia 
f2!""K  Key, 


\JX 


Local  Battery 


Relay 


Local  Battery 


Earth  Y\G.  287. 

line  wire  is  connected  with  the  earth  at  the  terminal  stations  by 
«ieans  of  metal  plates  sunk  in  moist  ground.  The  earth  com- 
pletes the  circuit,  taking  the  place  of  a  return  wire.  The  sounder 
at  each  station  is  in  a  local  circuit  connected  with  the  relay. 

When  an  operator  at  any  station  on  the  line  wishes  to  send  a 
message,  he  opens  the  switch  of  his  key.  All  other  switches  on 
the  line  must  be  closed.  (Why  ?)  The  operator  first  calls  the 
station  to  which  he  wishes  to  send  the  message.  The  sounders  at 
all  the  stations  deliver  the  message,  but  the  operator  at  the  station 
called  is  the  only  one  who  pays  attention  to  it. 

It  is  possible,  by  means  of  different  connections  and  instru- 
ments of  different  construction  from  those  here  described,  to  send 
two  or  more  messages  over  the  same  wire  at  the  same  time. 

IV.   Electrical  Measurements 

464.  Strength  of  Electric  Currents.  —  The  strength  of  an  electric 
current  can  be  determined  by  observing  how  great  an  effect  of  one 
kind  or  another  it  is  capable  of  producing.    Heating  and  chemical 

1  In  diagrams  each  cell  of  a  battery  is  represented  by  two  parallel  lines  (  1 1)  ;  the 
short  heavy  line  represents  the  negative  plate,  and  the  long,  thin  line  the  positive. 


370  Electricity 

effects  may  be  made  to  serve  this  purpose  ;  but  the  simplest  and 
most  convenient  method  is  to  use  some  form  of  instrument  whose 
action  is  due  to  the  magnetic  field  of  the  current.  Such  an  instru- 
ment is  called  a  galvanometer.  There  are  various  forms  of  galva- 
nometers, but  all  of  them  depend  in  their  action  upon  the  fact 
that  the  strength  of  an  electric  current  is  proportional  to  the  inten- 
sity of  its  magnetic  field. 

465.  Ohm's  Law.  —  Utilizing  the  magnetic  action  of  the  electric 
current  on  a  magnetic  needle,  (ieorg  Ohm,  a  German  physicist, 
discovered  that  the  strength  of  the  current  in  an  electric  circuit 
is  proportional  to  the  electro- motive  force  and  inversely  proportional 
to  the  resistance  of  the  circuit.  This  is  known  as  Ohm's  law.  It 
is  stated  in  other  terms  in  Art.  474. 

The  relations  included  in  the  law  may  be  separately  stated  as 
follows :  — 

(i)  If  the  E.  M.  F.  and  the  resistance  of  a  circuit  (including 
the  resistance  of  the  battery)  remain  constant,  the  current  remain^ 
constant. 

(2)  If  the  resistance  remains  constant  and  the  E.  M.  F.  varies, 
the  current  is  proportional  to  the  E.  M.  F. 

(3)  If  the  E.  M.  F.  remains  constant  and  the  resistance  varies, 
the  current  is  inversely  proportional  to  the  resistance.  For  exam- 
ple, if  the  resistance  is  doubled,  the  current  is  decreased  one  half. 

466.  The  Tangent  Galvanometer.  —  The  essential  parts  of  a 
tangent  galvanometer  (Fig.  288)  are  a  vertical 
coil  of  wire  having  one  or  more  turns,  and  a 
compass  with  a  graduated  scale,  placed  at  the 
center  of  the  coil.  The  compass  needle  should 
be  very  short  in  comparison  with  the  diameter 
of  the  coil,  in  order  that  it  may  be  wholly 
within  the  sensibly  constant  portion  of  the 
magnetic  field  of  the  current  at  and  near  the 

center  of  the  coil,  in  whatever  direction  it  may 
Fig.  288.  .    ,  .         .  r    y      ' 

turn.     A  long,  nonmagnetic  pomter  of  alumi- 
num is  commonly  attached  to  the  needle  at  right  angles  to  it 


Electrical  Measurements 


371 


Fig.  289. 


(Fig.  289)  ;  and  it  is  the  position  of  the  pointer  on  the  scale 
that  is  read.  It  is  evident  that  the  pointer  and  the  needle  turn 
through  equal  angles. 

At  and  near  the  center  of  the  coil  the  lines 
of  force  of  the  magnetic  field  of  the  current 
are  straight  and  are  perpendicular  to  the  plane 
of  the  coil  (Art.  455).  Hence,  if  the  current 
in  the  coil  were  the  only  source  of  a  magnetic 
field  where  the  compass  needle  is  placed,  the 
needle  would  be  brought  to  rest  exactly  at 
right  angles  to  the  plane  of  the  coil  whenever  a  current  of  any 
strength  was  flowing.  The  magnetic  field  of  the  earth,  however, 
plays  a  necessary  part  in  the  use  of  the  instrument,  the  behavior 
of  the  needle  being  determined  by  the  relative  intensity  of  the  two 
fields. 

.In  using  a  tangent  galvanometer  it  must  be  set  with  the  plane 
of  the  coil  in  the  magnetic  meridian ;  in  which  position  the  lines  of 
force  of  the  coil  are  at  right  angles  to  those  of  the  earth's  field. 
In  Fig.  290,  let  O  denote  the  position  of  the  north  pole  of  the 

compass  needle,  ON  the  intensity  and 
direction  of  the  earth's  magnetic  field, 
and  OD  the  intensity  and  direction 
of  the  field  of  the  current.  The  direc- 
tion of  the  resultant  of  these  forces  is 
OAj  which  is  therefore  the  direction  of  the  resultant  force  upon  the 
north  pole  of  the  needle.  Since  the  resultant  force  upon  the  south 
pole  is  equal  and  opposite  to  that  upon  the  north  pole,  the  needle 
comes  to  rest  in  the  line  OA,  and  the  deflection  of  the  needle 
caused  by  the  current  is  the  angle  NO  A, 

Any  increase  in  the  strength  of  the  current  through  the  galva- 
nometer increases  the  strength  of  its  field  i^OD)  proportionally,  and 
hence  increases  the  deflection,  but  not  proportionally.  This  is 
evident  from  the  two  parts  of  the  figure.  The  part  at  the  left 
represents  the  field  of  the  current  equal  to  that  of  the  earth,  and 
the  deflection  is  consequently  45°.     The  part  at  the  right  shows 


^f 


A   % 


Fig.  290. 


372 


Electricity 


V  A B 

O  J}  i' 


Fig.  991. 


the  effect  of  doubling  the  current  (which  makes  OD  twice  as 
great) ;  the  deflection  is  increased,  but  is  much  less  than  90°.  In 
fact,  the  deflection  is  never  quite  90°,  how- 
ever great  the  current  may  be. 

Let  C  and  C  denote  two  currents 
through  the  same  number  of  turns  of  the 
same  galvanometer,  and  let  OD  and  0£, 
or  JVA  and  NB  (Fig.  291),  denote  the 
intensities  of  their  fields  at  the  center  of 
the  .galvanometer.  OJV^,  as  before,  is  the  intensity  of  the  earth's 
field.  Then  angles  a  and  a'  are  the  deflections  caused  by  C  and 
C  respectively. 

Now  the  strengths  of  the  currents  are  proportional  to  the  inten- 
sities of  their  magnetic  fields  (Art.  464),  i>. 

C:  C::NA:NB, 

Since  the  value  of  a  ratio  is  not  altered  when  its  terms  are 
divided  by  the  same  quantity,  we  may  write  the  above  proportion 


C:  C: 


NA  ,  NB 
ON  '  ON 


(I) 


The  ratio  of  one  leg  of  a  right  triangle  to  the  other  is  called  the 

NA 
tangent^  of  the  angle  opposite  to  the  first  leg.     Thus is  the 

tangent  of  angle  a  (commonly  abbreviated  to  tan  tf),  and is 


1  The  tangent  of  angle  A  (Fig.  a^a)  is  DE  :  AD  or  PG  :  AG,  DE  and  FG  being 
any  line  perpendicular  to  either  side  of  the  given  angle.  Since  triangles  ADE  and 
AFG  are  similar,  the  ratios  DE :  AD  and  EG : 
AG  are  equal.  It  is  evident,  therefore,  that 
the  tangent  of  an  angle  is  a  definite  quantity 
the  value  of  which  depends  only  upon  the  sire 
of  the  angle.  Angles  are  not  prop>ortional  to 
their  tangents,  although  small  angles  are  very 
nearly  so.  The  tangent  of  any  angle  from  o''  to 
90^*  may  be  found  from  a  table  of  tangents 
(Appendix,  Table  V). 


Electrical  Measurements  373 

the  tangent  of  angle  a!  {tan  a').     Hence  proportion  (i)  may  be 

written  /->    r^t      4.  4.        t  /  \ 

C  :  C  :  :  tan  a  :  tan  a'.  (2) 

That  is,  currents  sent  through  the  same  number  of  turns  of  the 
coil  of  a  tangent  galvanometer  are  proportional  to  the  tangents  of 
the  angles  of  deflection  that  they  produce.  This  is  why  the  instru- 
ment is  called  a  tangent  galvanometer. 

Example.  —  A  current  C  causes  a  deflection  of  50°,  and  another  current  C, 
a  deflection  of  25*^.  It  is  found  from  a  table  of  tangents  that  tan  50°  =1.19 
and  tan  25°  =  .466. 

Hence  C:  C: :  1.19:  .466  ; 

from  which  C  —  2.55  C, 

While  a  tangent  galvanometer  having  a  scale  graduated  in  de- 
grees may  be  thus  used  to  determine  the  relative  strengths  of 
currents,  it  does  not  give'  their  numerical  values.  The  numerical 
value  (in  amperes)  may,  however,  be  obtained  by  multiplying  the 
tangent  of  the  angle  of  deflection  by  a  constant  factor,  found  by 
experiment. 

Laboratory  Exercise  68. 

467.  Use  of  Different  Numbers  of  Turns  of  the  Coil.  —  When 
equal  currents  are  sent  through  different  numbers  of  turns  of  a 
tangent  galvanometer,  the  tangents  of  the  angles  of  deflection  are 
proportional  to  the  number  of  turns  used  (Lab.  Ex. -68).  For 
example,  when  a  current  is  passed  through  fifteen  turns,  the  tan- 
gent of  the  angle  of  deflection  is  three  times  as  great  as  when  the 
same  current  is  sent  through  five  turns.  This  is  due  to  the  fact 
that  the  magnetic  field  of  the  coil  is  proportional  to  the  number 
of  turns  through  which  the  current  is  sent  (see  note  to  Art.  454). 
In  measuring  currents  the  most  accurate  results  are  obtained  when 
the  number  of  turns  used  is  the  one  that  makes  the  deflection 
nearest  to  45°. 

PROBLEMS 

1.  How  would  the  current  from  a  given  battery  be  affected  by  making  the 
resistance  of  the  circuit  four  times  as  great  ?  ten  times  as  great  ? 

2.  Make  a  diagram  showing  the  deflection   of  the  needle  of  a  tangent 


374  Electricity 

galvanometer  due  to  a  certain  current,  and  also  the  deflection  due  ta 
currents  two,  three,  four,  and  five  times  as  great.  What  change  in  the 
increase  of  the  deflection  for  successive  equal  increases  of  the  current  is 
shown  by  the  diagram  ? 

3.  What  value  docs  the  deflection  approach  as  the  current  increases 
indefmitely  ? 

4.  Why  docs  the  stronger  or  weaker  magnetization  of  a  galvanometer 
needle  not  atfect  the  deflection  ? 

Laboratory  Exercise  6g. 

468.  Laws  of  Resistance.  —  The  following  laws  of  electrical 
resistance  have  been  established  by  experiment :  — 

I.  Thf  resistance  of  a  conductor  of  uniform  cross-section  is  pro- 
portional to  its  length. 

II.  The  resistance  of  a  conductor  is  inversely  proportional  to  the 
area  of  its  cross-section.  The  resistance  of  a  wire  is,  therefore, 
inversely  proportional  to  the  square  of  its  diameter.  For  example, 
the  resistance  of  a  wire  3  mm.  in  diameter  is  one  ninth  as  great  as 
that  of  a  wire  of  the  same  material  and  length  and  i  mm.  in 
diameter. 

IIL  The  resistance  of  a  conductor  depends  upon  the  material  of 
which  it  is  made.  The  following  table  gives  the  relative  or  specific 
resistances  of  a  number  of  substances  in  the  form  of  wires  of  equal 
length  and  cross- section,  the  resistance  of  copper  being  taken  as 
unity. 

Specific  Resistances  (referred  to  copper) 

Silver,  annealed 0.94  Iron,  telegraph  wire    .     .     .  9.4 

Copper,  annealed i.oo  German  silver 13.0 

Aluminum 1. 81  Mercury 59.0 

Iron,  pure 6.03  Carbon  (arc  light)  about .     .  2500.0 

IV.  The  resistance  of  metals  and  of  mqst  other  substances  in- 
creases as  the  temperature  rises.  TJie  resistance  of  carbon^  dilute 
acidSf  and  solutions  decreases  with  a  rise  of  temperature.  The 
resistance  of  nearly  all  the  pure  metals  increases  about  40  per 
cent  with  a  rise  of  temperature  of  100°  C.  The  resistance  of 
German  silver  and  other  alloys  is  much  less  affected  by  change  of 


Electrical  Measurements 


375 


temperature  ;  hence  they  are  used  in  making  standard  resistance 
coils  (Art.  470).  The  resistance  of  the  carbon  filament  of  an 
incandescent  lamp  when  hot  is  only  about  half  what  it  is  when 
cold. 

V.  The  resistance  of  a  conductor  is  the  same  at  the  same  tem- 
perature whatever  the  strength  of  the  current. 

469.  The  Unit  of  Resistance.  —  The  common  unit  of  electrical 
resistance  is  called  the  ohm^  in  honor  of  the  physicist  of  that 
name.  It  is  defined  as  the  resistance  of  a  uniform  column  of  mer- 
cury 106.3  cm.  long  and  one  square  millimeter  in  cross-section, 
at  0°  C.  It  is  very  approximately  the  resistance  of  157  ft.  of 
No.  18  copper  wire  (diameter  =  1.024  mm-)  or  349  ft.  of  No.  16 
(diameter  =  1.29  mm.). 

470.  Resistance  Coils.  —  Standard  coils  of  known  resistance  are 
used  in  measuring  resistances.  They  are  called  resistance  coils, 
and  a  box  containing  a  set 
of  them  is  called  a  resist- 
ance box  (Fig.  293).  The 
coils  are  made  of  insulated 
wire,  generally  of  an  alloy 
having  a  high  specific  re- 
sistance j  and  the  ends  of 
each  coil  are  connected  to 
brass  blocks,  A,  By  C 
(Fig.  294),  on  the  top  of 
the  box.  These  blocks  are  separated  a  short  distance ;  but  they 
are  connected  electrically  by  means  of  the  coils,  and  may  also 

be  connected  by  inserting  brass  plugs,  which 
fit  snugly  into  the  spaces  between  them.  The 
resistance  of  the  row  of  blocks  and  plugs  is 
practically  zero ;  but,  wherever  a  plug  is  re- 
moved, the  resistance  of  the  coil  that  bridges 
the  gap  is  introduced  into  the  circuit  in  which 
The  amount  of  this  resistance  is  marked 
When  two  or  more  plugs  are  removed, 


^93- 


Fig.  294. 

the  box  is  included, 
on  the  top  of  the  box. 


3/6  Electricity 

the  total  resistance  introduced  is  the  sum  of  the  resistances  of 
the  coils  that  connect  the  gaps.  A  set  of  coils  generally  includes 
.1,  .2,  .3,  .4,  I,  2,  3,  4,  10,  20,  30,  and  40  ohms,  and  may  extend 
to  much  higher  resistances.  Any  resistance  from  .1  ohm  to 
III  ohms  can  be  introduced  into  a  circuit  with  such  a  box.  In 
finding  a  required  resistance,  the  coils  are  tried  in  order  from  larger 
to  smaller,  as  weights  are  tried  in  weighing. 

471.  Measurement  of  Resistance.  —  The  resistance  of  a  con- 
ductor can  be  measured  in  a  number  of  ways ;  but  the  method 
of  substitution  is  the  only  one  that  we  shall  consider.  Suppose 
the  resistance  of  a  coil  of  wire,  R  (Fig.  295), 
j\->.   is  to  be  found.     A  constant  cell  (Art  451)  is 


connected  with  a  galvanometer,  <?,  and  the 
coil  is  included  in  the  circuit.  The  deflection 
"*  of  the  galvanometer  is  read  as  accurately  as 

*  possible.    The  coil  is  then  removed  from  the 

circuit  and  a  resistance  box  put  in  its  place.  Different  resist- 
ances are  introduced  into  the  circuit,  by  removing  plugs  from  the 
box,  till  the  deflection  of  the  galvanometer  is  exactly  the  same  as 
with  the  coil,  R. 

The  sum  of  the  resistances  of  the  box  then  included  in  the 
circuit  is  equal  to  the  resistance  of  the  coil,  R.  The  proof  of  this 
is  as  follows :  ( i )  The  equal  deflections  indicate  equal  currents 
(formula  (2),  Art.  466).  (2)  The  E.  M.  F.  of  the  cell  is  constant ; 
hence  the  current  is  inversely  proportional  to  the  resistance  of  the 
circuit  (Art  465,  (3)),  and,  as  the  currents  are  known  to  be 
equal,  the  resistances  of  the  circuits  must  be  equal.  (3)  Hence 
the  resistances  introduced  from  the  box  must,  be  equal  to  the 
resistance  of  R,  for  which  they  were  substituted. 

Laboratory  Exercises  yo  and  yi. 

472.  The  Unit  of  Electro-motive  Force.  —  The  unit  of  electro- 
motive force  is  called  the  volt,  in  honor  of  Volta  (Art.  442).  It  is 
defined  as  a  certain  fraction  of  the  E.  M.  F.  of  a  Clark  standard 
cell  at  15°.  This  cell  is  chosen  as  the  standard  because  of  its 
constancy.    The  E.  M.  F.  of  a  gravity  or  a  Daniell  cell  is  approxi- 


Electrical    Measurements  377 

mately  i  volt ;  of  a  chromic  acid  cell,  2  volts ;  of  a  Leclanch^ 
cell,  1.4  volts.  The  E.  M.  F.  of  a  cell  is  independent  of  its  size 
and  dimensions,  but  varies  more  or  less  with  the  condition  of  the 
plates  and  the  liquid. 

473.  The  Unit  of  Current.  —  The  unit  of  current  is  called  the 
ampere  J  in  honor  of  Andre  Ampere,  a  French  physicist,  noted  for 
his  discoveries  in  electro-magnetism.  It  may  be  defined  indepen- 
dently of  the  ohm  and  the  volt,  in  terms  of  either  its  magnetic  or  its 
chemical  effects ;  but  it  will  serve  our  purpose  to  define  it  as  the 
current  produced  by  an  E.  M.  F.  of  one  volt  in  a  conductor  having 
a  resistance  of  one  ohm. 

474.  Ohm's  Law.  —  The  units  of  current,  E.  M.  F.,  and  resist- 
ance being  thus  chosen  with  reference  to  one  another,  Ohm's 
law  is  expressed  by  the  formula 

c=|.  (0 

in  which  C  denotes  the  current  measured  in  amperes^  E  the 
E.  M.  F.  measured  in  voltSy  and  R  the  resistance  of  the  whole 
circuit  measured  in  ohms.  From  the  formula,  when  E  and  R  are 
each  one,  C  is  one  ;  which  agrees  with  the  above  definition  of  the 
unit  current. 

475.  Fall  of  Potential  along  a  Conductor.  —  It  can  be  shown 
experimentally  that  there  is  a  continuous  fall 

of  potential  round  a  circuit  from  the  positive     ^ iL 

pole  of  the  battery  to  the  negative  pole,  and     ^  ' 

that  the  fall  of  potential  is  everywhere  pro- 
portional to  the  resistance  to  be  overcome. 
Thus  if  E^  denote  the  fall  of  potential  or  ^ 

E.M.F.  between  A  and  B  (Fig.  296)  and  ^'°*  '^ 

E^  the  E.  M.  F.  between  B  and  C,  and  if  Ry  and  R^  denote  respec- 
tively the  resistance  oi  AB  and  BCy  then 

E^:E,:'.R,:R,,  or  |l  =  ^.  (2) 

Ri      Aj 


3/8  Electricity 

Let  C  denote  the  current  (which  is  the  same  throughout  the 
circuit),  then 

C  =  |  and  C  =  |^  (3) 

These  equations  express  the  fact  that  Ohm's  law  holds  for  parts 
of  circuits  as  well  as  for  entire  circuits. 

476.  Divided  Circuits.  —  Electric  currents  often  have  two  or 
more  branches  between  two  points,  as  between  A  and  B  (Fig. 

297).  Either  of  two  branches  is  called  a 
shunt  to  the  other.  The  sum  of  the  currents 
in  all  the  branches  between  two  points  is 
equal  to  the  current  in  the  undivided  part 
of  the  circuit 

In  Fig.  297  let  ^1  denote  the  resistance 
of  one  branch  of  the  circuit  between  A  and 
B,  and  ^,  the  resistance  of  the  other ;  and  let  Ci  and  Q  denote 
the  currents  in  these  branches  respectively.  The  E.  M.  F.  that 
maintains  the  current  in  each  branch  is  the  difference  of  potential 
between  the  points  A  and  B,  Let  E  denote  this  E.  M.  F. ;  then, 
by  Ohm's  law, 

E=  C,^i,  and  E=  C^j. 
Hence,  C^R^  =  C^R^  or  Q:  Q::  R^:  Ri;  (4) 

that  is,  the  currents  in  the  two  branches  are  inversely  proportional 
to  the  resistances  of  the  branches.  For  example,  if  the  resistance 
of  one  branch  is  three  times  that  of  the  other,  the  current  in  it  will 
be  one  third  as  great  as  in  the  other,  or  it  will  carry  one  fourth 
of  the  whole  current. 

477.  Resistance  of  Conductors  in  Series  and  in  Parallel.  —  Two 
or  more  conductors  are  said  to  be  in  series  when  they  are  con- 
nected so  that  the  entire  current  passes  through  each  in  succession. 
Thus,  in  Fig.  296,  AB  and  BC  are  in  series.  The  combined 
resistance  of  any  number  of  conductors  in  series  is  the  sum  of 
their  separate  resistances. 

The  conductors  constituting  the  branches  of  a  divided  circuit 


Electrical  Measurements  379 

are  said  to  be  connected  ///  parallel.  The  resistance  between  any 
two  points  of  a  circuit,  as  between  A  and  B  of  Fig.  297,  is  always 
diminislied  by  adding  one  or  more  conductors  in  parallel  between 
those  points;  for  this  is  equivalent  to  replacing  the  given  con- 
ductor with  one  of  larger  cross-section.  For  example,  in  the 
simple  case  where  the  branches  are  of  the  same  material,  length, 
and  cross-section,  the  resistance  of  three  of  them  in  parallel  is 
one  third  the  resistance  of  one  of  them,  —  being  the  same  as  the 
resistance  of  a  single  conductor  of  the  same  material  and  length 
and  three  times  the  cross-section  (Art.  468,  Law  II). 

478.  Measurement  of  E.  M.  F. ;  The  Voltmeter.  —  The  coils  of 
some  galvanometers  consist  of  a  great  many  turns  of  very  fine  wire, 
and  have  a  resistance  of  hundreds  or  even  thousands  of  ohms. 
On  account  of  the  large  number  of  turns  in  the  coil  a  very  weak 
current — often  less  than  a  thousandth  of  an  ampere  —  causes  a 
considerable  deflection  of  the  needle.  Such  instruments  are  used 
to  determine  the  difference  of  potential,  or  E.  M.  F.,  between 
different  points  of  a  circuit,  and  are  called  voltmeters.  An 
E.  M.  F.,  when  measured  in  volts,  is  often  called  voltage. 

To  determine  the  E.  M.  F.  between  two  points  of  a  circuit,  as 
A  and  B  (Fig.  298),  the  voltmeter,  V^  is 
connected  as  a  shunt  between  these  points. 
The  resistance  of  the  voltmeter  must  be  very 
high  in  comparison  with  that  of  the  circuit 
between  A  and  B^  in  order  that  it  may  not 
appreciably  diminish  the  resistance  of  the 
circuit  between  these  points.  This  is  neces- 
sary ;  for  if  the  resistance  of  the  voltmeter 
were  low,  it  would  lower  the  resistance  between  the  points  to 
which  it  was  connected,  and  hence  would  diminish  the  E.  M.  F. 
between  these  points  in  the  act  of  measuring  it  (Art.  475). 

By  Ohm's  law,  the  current  through  the  voltmeter  is  proportional 
to  the  E.  M.  F.  between  the  points  of  the  circuit  to  which  it  is 
connected.  Hence,  if  it  is  a  tangent  instrument,  the  E.  M.  F. 
is  proportional  to  the   tangent  of  the  angle  of  deflection.     The 


380  Electricity 

scale  may  be  marked  in  degrees  or  graduated  so  as  to  give  the 
number  of  volts  directly.     To  find  the  total  E.  M.  F.  of  a  battery, 
its  poles  are  connected  directly  with  a  voltmeter  (Fig.  299). 
Laboratory  Exercise  72. 

479.  Arrangement  of  Cells  in  a  Battery.  —  The  cells  of  a 
battery  may  be  connected  in  three  ways,  namely:  (i)  in  series ^ 
(2)  in  parallel^  (3)  in  series  and  parallel  combined.  Which  of 
these  arrangements  will  give  the  largest  current  through  a  given 
circuit  depends  upon  the  relative  resistance  of  the  battery  and  the 
remainder  of  the  circuit,  as  shown  in  the  following  articles.  In 
this  discussion  E  denotes  the  E.  M.  F.  and  r  the  resistance  of 
each  cell,  n  the  number  of  cells,  and  R  the  external  resistance  ;  i.e, 
the  resistance  of  the  circuit  exclusive  of  the  battery  resistance. 
Ohm's  law,  when  applied  to  a  circuit  containing  only  one  cell, 
would  then  be  represented  by  the  formula 

r-\-  R  being  the  total  resistance  of  the  circuit. 

480.  Cells  in  Series.  —  Cells  are  said  to  be  connected  in  series 
when  the  positive  pole  of  the  first  is  joined  to  the  negative  pole 
of  the  second,  the  positive  pole  of  the  second  to  the  negative 

I  I  I  I        pole  of  the  third,  etc.  (Fig.  299).     The 

r\         'I        ^1        ^K     fall  of  potential  in  the  short  wires  con- 

V__,^____/T\ J    necting  the  cells  is  negligible  (their  re- 

^-^►^  sistance    being  very  small),   hence   the 

Fig.  299.  negative  plate  of  each  cell  is  at  the  same 

potential  as  the  positive  plate  of  the  preceding  cell ;  and,  since 
the  rise  of  potential  in  each  cell  is  E^  the  E.  M.  F.  of  a  battery 
of  n  cells  in  series  is  nE,  The  resistance  of  the  battery  is  the 
sum  of  the  resistances  of  the  cells,  or  «r,  according  to  the  law 
for  conductors  in  series  (Art.  477).  Ohm's  law,  when  applied  to 
a  circuit  containing  a  battery  of  n  like  cells  in  series,  is  therefore 
represented  by  the  formula 

C=-^'  (6) 

nr  +  R  ^ 


Electrical  Measurements 


381 


When  the  external  resistance,  R^  is  small  in  comparison  with 
the  battery  resistance,  nr,  the  current  is  only  very  slightly  increased 
by  adding  more  cells  in  series,  for  this  increases  the  resistance  of 
the  circuit  in  nearly  the  same  ratio  as  it  does  the  E.  M.  F. ;  but, 
when  the  external  resistance  is  relatively  large,  an  increase  in  the 
number  of  cells  increases  the  E.  M.  F.  more  rapidly  than  it  does 
the  total  resistance,  and  hence  increases  the  current. 

481.   Cells  in  Parallel.  —  Cells  are  said  to  be  connected  in 
parallel  when  the  negative  plates  of  all  the  cells  are  joined  to  form 
the  negative  pole  of  the  battery  and       __,^_ 
the  positive  plates  to  form  the  positive    -^  '  ^     ^:-  ^-^     © 
pole   (Fig.    300).     All  the  negative 
plates  are  at  one  potential  and  all 

the  positive  plates  at  another,  the  difference  being  the  E.  M.  F. 
of  one  cell.     The  resistance  of  a  battery  of  n  cells  in  parallel  is 

I  T 

-  of  the  resistance  of  a  single  cell,  or  -,  according  to  the  law 

n 

Hence  the  formula  for  Ohm's  law 
C=^-  (7) 


for  conductors  in  parallel, 
in  this  case  is 


n 
When  the  resistance  of  a  single  cell  is  small  compared  with  the 
external  resistance,  the  current  is  only  slightly  increased  by  con- 
necting any  number  of  cells  in  parallel,  for  the  total  resistance 
is  only  slightly  diminished ;  but,  when  the  external  resistance  is 

relatively  small,  any  de- 
crease in  the  battery 
resistance  decreases  the 
total  resistance  in  nearly 
the  same  ratio  and  pro- 
duces a  corresponding 
increase  in  the  current. 
482.  Cells  in  Series 
and  Parallel  Combined.  —  In  some  cases  the  largest  current  is 
obtained  by  arranging  the  cells  of  a  battery  in  groups.     The  result 


Fig.  301. 


Fig.  302. 


382  Electricity 

is  the  same  whether  the  cells  of  each  group  are  joined  in  series, 
and  the  groups  in  parallel  (Fig.  301),  or  the  cells  of  each  group 
in  parallel  and  the  groups  in  series  (Fig.  302).  The  formula  for 
the  arrangement  shown  in  either  figure  is 

3-:+/. 

2 

It  can  be  shown  mathematically  that,  with  a  given  number  of 
cells,  the  largest  current  through  a  given  external  resistance  is 
obtained  when  the  cells  are  so  connected  that  the  resistance  of  the 
battery  is  as  nearly  as  possible  equal  to  the  external  resistance. 

Laboratory  Exercise  jj. 

PROBLEMS 

1.  What  is  the  combined  resistance  of  three  incandescent  lamps  in 
parallel,  the  resbtance  of  each  lamp  being  2CX)  ohms? 

2.  The  fall  of  potential  through  a  coil  of  wire  is  1.5  volts  when  a  current 
of  .2  ampere  is  flowing.     What  is  the  resistance  of  the  coil  ? 

3.  What  E.  M.F.  will  maintain  a  current  of  1.5  amperes  through  a  re- 
sistance of  80  ohms? 

4.  If  the  E.  M.  F.  of  a  chromic  acid  cell  is  2  volts,  and  its  resistance 
.3  ohm,  what  current  will  it  supply  through  an  external  resistance  of  .1  ohm? 

5.  What  would  be  the  current  through  the  same  external  resistance  from 
a  battery  of  4  such  cells  (a)  in  parallel?  {b)  in  series? 

6.  \\'hat  current  would  be  supplied  by  a  battery  of  12  Leclanche  cells, 
each  having  an  E.  M.  F.  of  14  volts  and  a  resistance  of  i  ohm,  through  an 
external  resistance  of  I.5  ohms  {a)  with  the  cells  connected  in  series?  {b)  in 
parallel?  (<-)  in  three  groups  of  four  each,  the  cells  of  each  group  being  in 
series,  and  the  groups  connected  in  parallel?  Draw  a  diagram  of  the  last 
arrangement 

7.  Show  that,  when  the  external  resistance  of  a  circuit  is  negligible  in 
comparison  with  the  resistance  of  a  cell,  the  current  is  proportional  to  the 
number  of  cells  connected  in  parallel,  but  a  single  cell  furnishes  as  large  a 
current  as  any  number  of  cells  connected  in  series. 

8.  Show  that,  when  the  battery  resistance  is  negligible  in  comparison 
with  the  external  resistance,  the  current  is  proportional  to  the  number  of 
cells  connected  in  series,  but  a  single  cell  furnishes  as  large  a  current  as  any 
number  of  cells  in  parallel. 


Electrical  Measurements 


383 


483.  The  Astatic  Galvanometer.  —  This  instrument  (Fig.  303) 
is  used  for  detecting  and  measuring  extremely  small  currents.     It 
has  an  astatic  needle  (Fig.  304).     This  con- 
sists of  two  magnetic  needles  placed  parallel, 
one  above  the  other,  with  their  like  poles  point- 
ing in  opposite  directions.     They  are  rigidly 
attached  to  a  short,  vertical  stem,  and  hence 
turn  together.    If  they  were  of  exacdy  the  same 
strength,  they  would  come  to  rest  indifferently 
in  any  position,  for  the  earth's  magnetic  field 
would  act  upon  them  equally  and  in  opposite 
directions ;  but  a  very  slight  directive  action  is  ^'^'  3^* 
secured  by  making  one  needle  a  little  stronger  than  the  other. 
The  needle  carries  a  light,  nonmagnetic  pointer,  and  is  suspended 

by  an  untwisted  silk  fij^er,  which  en- 
ables it  to  respond  to  the  slightest 
force. 

The  coil  of  the  instrument  is  flat 
and  elongated  horizontally.  The  lower 
needle  swings  within  the  coil,  the 
upper  one  above  it.  The  magnetic 
field  of  the  coil  tends  to  turn  both  needles  in  tne  same  direction. 
(Why?)  Thus  with  the  astatic  needle  the  effect  of  the  current  is 
increased  while  that  of  the  earth's  field  is  decreased,  and  a  very 
small  current  causes  a  considerable  deflection.  The  astatic  galva- 
nometer is  therefore  called  a  sensitive  instrument. 

484.  The  d'Arsonval  Galvanometer.  —  This  galvanometer  (Fig. 
305)  differs  in  principle  from  those  previously  considered.  A  coil 
of  fine  wire,  wound  on  an  elongated  frame,  is  suspended  so  as  to 
swing  freely  between  the  poles  of  a  permanent  horseslioe  magnet. 
The  current  is  led  to  and  from  the  coil  through  wires  by  means  of 
which  it  is  suspended.  When  no  current  is  flowing,  the  suspending 
wires  hold  the  coil  in  position  with  its  axis  at  right  angles  to  the 
lines  of  force  between  the  poles  of  the  magnet ;  but,  with  a  current 
flowing,  the  coil  tends  to  set  itself  with  its  north  side  facing  the 


Fig.  304. 


384 


Electricity 


Fid. 


305. 


south  pole  of  the  magnet  (Art.  456).  This  motion  is  opposed  by 
the  torsion  of  the  wire,  and  the  coil  turns  through  a  greater  or  less 
angle  according  to  the  strength  of  the  current.  In  some  instru- 
ments the  coil  is  provided  with 
a  nonmagnetic  pointer,  which 
moves  over  a  scale,  as  in  the 
figure  ;  in  others  the  coil  carries 
a  small  mirror,  which  indicates 
the  deflection  by  the  angle  at 
which  it  reflects  a  beam  of  light. 
One  important  advantage  of 
the  d'Arsonval  galvanometer  is 
that  it  is  independent  of  the 
earth's  field,  which  is  negligible 
in  comparison  with  the  strong 
field  of  the  magnet ;  hence  the 
instrument  does  not  need  to  be  turned  in  any  particular  direction. 
The  sensitiveness  of  this  galvanometer  is  increased  by  decreasing 
the  size  of  the  supporting  wires,  by  increasing  the  strength  of  the 
magnet,  or  by  increasing  the  number  of  turns  of  the  coil.  A 
sensitiveness  sufficient  to  detect  much  less  than  a  millionth  of  an 
ampere  can  be  secured. 

485.  The  Weston  Ammeter  and  Voltmeter.  —  A  galvanometer 
whose  scale  is  graduated  to  read  in  amperes  is  called  an  ammeter. 
Ammeters  and  voltmeters  for  indus- 
trial use  are  usually  of  the  d'Arsonval 
type,  and  resemble  each  other  in 
appearance.  Figure  306  represents 
a  Weston  ammeter.  The  magnet  is 
horizontal,  its  poles  being  at  the  nar- 
row side  of  the  box,  opposite  the  scale. 
The  magnetic  force  acting  upon  the 
coil  is  opposed  by  a  coiled  spring. 
The  coil  carries  a  long  pointer 
which  moves  over  a  scale  graduated  in  amperes.  In  the  ammeter 
the  resistance  of  the  coil  is  low ;  in  the  voltmeter  it  is  very  high. 


Fig.  306. 


Heat,  Light,  and  Power  385 

V.   Heat,  Light,  and  Power  from  Electric  Currents 

486.  Heating  Effect  of  Electric  Currents.  —  When  the  current 
from  a  chromic  acid  battery  is  sent  through  a  short  piece  of  fine 
German  silver  or  platinum  wire,  the  wire  becomes  white  hot ;  but 
the  remainder  of  the  circuit,  consisting  of  larger  copper  wire,  is 
not  appreciably  warmed  (Exp.). 

The  greater  heating  of  the  German  silver  or  platinum  wire  is 
due  to  its  greater  resistance.  A  part  of  the  energy  of  an  electric 
current  is  transformed  into  heat  wherever  there  is  resistance  to  be 
overcome,  whether  in  the  battery  or  in  the  external  circuit ;  and 
it  is  found  by  experiment  that  the  amount  of  heat  thus  generated 
in  the  different  parts  of  the  circuit  is  directly  proportional  to  their 
resistances.  Heat  is  generated  in  a  conductor  as  long  as  the  cur- 
rent flows ;  but  the  temperature  of  the  conductor  ceases  to  rise  as 
soon  as  the  rate  at  which  it  loses  heat  by  conduction  and  radiation 
becomes  equal  to  the  rate  at  which  the  heat  is  generated. 

487.  The  Incandescent  Lamp.  —  The  heating  of  electric  cur- 
rents is  utilized  in  electric  lighting.  In  the  incan- 
descent lamp  (Fig.  307)  the  current  passes  through 
a  slender  carbon  filament  which,  on  account  of  its 
high  resistance,  is  heated  to  whiteness.  The  ends 
of  the  filament  are  attached  to  short  platinum 
wires,  which  pass  through  the  glass  and  make  con- 
nection with  the  circuit  by  means  of  the  metal  fittings 
at  the  base  of  the  lamp.  The  bulb  is  exhausted  to 
the  highest  possible  degree  to  prevent  combustion 
of  the  filament  when  heated.  The  common  sixteen- 
candle  lamp  has  a  resistance  of  160  to  200  ohms  yig.  307. 
when  hot,  and  requires  a  potential  difference  of  100 

or  no  volts  between  its  terminals  to  maintain  the  necessary  cur- 
rent, which  is  about  .5  to  .6  ampere.  The  lamps  are  commonly 
connected  in  parallel. 

488.  The  Electric  Arc.  —  If  two  pointed  pieces  of  carbon  are 
placed  in  contact,  then  separated  a  short  distance  while  a  current 


386 


Electricity 


Fig.  308. 


of  sufficient  strength  and  voltage  is  passing  between  them,  the 
heat  generated  at  the  points  will  vaporize  some  of  the  carbon, 
and  the  current  will  continue  to  flow,  being 
conducted  across  the  gap  between  the  points 
by  the  carbon  vapor  and  heated  air  (Fig. 
308).  The  conducting  gases  form  a  curved 
luminous  path,  which  is  known  as  the  electric 
or  voltaic  arc.  The  greater  part  of  the  light 
comes  from  the  carbon  points,  especially  from 
the  depression  or  crater  which  forms  at  the 
end  of  the  positive  carbon. 

The  light  of  the  electric  arc  is  the  most 
powerful  and  its  temperature  the  highest  that 
can  be  artificially  produced.  The  tempera- 
ture is  estimated  at  from  3500"  to  4800°  C, 
and  is  sufficient  to  melt  the  most  infusible 
substances,  such  as  flint  and  diamond.  An  Y..  M.  F.  of  40  to  50 
volts  and  a  current  of  from  5  to  10  amperes  are  required  to 
produce  a  steady  arc  light ;  and  such  a  light  will  have  from  1000 
to  2000  candle  power.  The  currents  for  both  arc  and  incandes- 
cent lights  may  be  either  direct  or  alternating  and  are  supplied  by 
dynamos. 

Arc  lamps  are  provided  with  an  auto- 
matic device  controlled  by  electro-mag- 
nets, by  means  of  which  the  carbons  are 
brought  in  contact  when  the  current  is 
not  flowing  and  separated  as  soon  as  it 
is  turned  on,  the  action  of  the  electro- 
magnets being  controlled  by  the  current. 
This  device  also  "feeds"  the  upper 
carbon  toward  the  lower  as  fast  as  they 
are  consumed. 

489.  Joule's  Law.  —  Dr.  Joule  (Art.  270)  investigated  the 
heating  effect  of  electric  currents  by  passing  a  current  through  a 
coil  of  wire  of  known  resistance  in  a  calorimeter  containing  water 


Fig.  309, 


Heat,   Light,  and   Power  387 

or  alcohol  (Fig.  309).  From  the  results  thus  obtained  he  found 
that  the  heat  developed  in  a  conductor  is  proportional  {i)  fo  the 
resistance  of  the  conductor^  (2)  /'^  the  time  during  which  the  cur- 
rent flows  ^  and  (3)  to  the  square  of  the  strength  of  the  current. 
These  relations  are  expressed  in  the  equation 

H=.2^C^Rt,  (8) 

in  which  H  denotes  the   number  of  calories   generated,   C  the 
current  in  amperes,  R  the  resistance  of  the  conductor  in  ohms, 
and  /  the  time  in  seconds.     This  equation  is  known  2isfoule's  law. 
Since  by  Ohm's  law  E  =  CR^  Joule's  law  may  be  written 

H=-.2^ECt,  (9) 

in  which  E  denotes  the  difference  of  potential  between  the  ends 
of  the  conductor  in  which  the  heat  is  generated. 

490.  Electrical  Energy.  —  'Hie  heat  generated  in  a  conductor 
through  which  a  current  is  flowing  is  the  equivalent  of  the  energy 
lost  by  the  current  in  overcoming  the  resistance  of  the  conductor.. 
This  loss  of  energy  is  accompanied  by  a  fall  of  potential,  but  the 
current  is  not  diminished  —  the  reading  of  an  ammeter  is  the  same 
at  whatever  point  in  the  circuit  it  may  be  placed.  It  is  evident 
from  this  that  electrical  energy  is  not  electricity ^  any  more  than  the 
energy  of  a  stream  of  water  is  the  water  itself.  Water  falls  to  a 
lower  level,  and  hence  loses  potential  energy,  in  turning  a  mill 
wheel ;  but  there  is  no  loss  of  water  in  the  process.  Similarly 
there  is  a  fall  of  electrical  potential  (as  may  be  shown  by  a  volt- 
meter) whenever  an  electric  current  does  work,  whether  it  be  in 
overcoming  the  resistance  of  the  conductor,  in  running  a  motor 
(Art.  505),  or  in  doing  chemical  work  (Art.  511). 

///  all  cases  the  energy  lost  by  the  current,  or  the  work  done  by  it, 
is  proportional  Jointly  (i)  to  the  fall  of  potential,  (2)  to  the  current, 
and  (3)  to  the  time  ;  i.e.  the  work  done  is  proportional  to  the  prod- 
uct ECt, 

491.  Power.  —  The,  work  done  by  an  electric  current  in  a 
second  is  called  \is  power  (Art.  153).  Since  the  work  done  by 
a  current  is  proportional  to    ECt,  the  work  done  in  a  second, 


388  Electricity 

or  the  power,  is  proportional  to  ^C;  i>.  stated  more  fully,  the 
power  of  a  current  utilized  in  any  part  of  the  circuit  is  pro- 
portional to  the  strength  of  the  current  and  to  the  fall  of  potential 
in  that  part  of  the  circuit.  The  electric  unit  of  power  is  called 
the  watt;  it  is  defined  as  the  power  of  a  current  of  one  ampere 
when  driven  by  an  E.  M.  F.  of  one  volt.  Hence,  the  power  of  a 
current  in  watts  is  equal  to  the  product  of  the  volts  and  the 
amperes;  that  is, 

Power  ^  EC  watts.  (10) 

One  thousand  watts  is  called  a  kiloioatt;  hence, 

EC 

Power  = kilowatts.  (11) 

1000 

It  can  be  shown  that  one  horse  i)ower  (Art.  153)  is  equal  to  746 

watts;  hence,  ^q 

Power  =  —  horse  power.  (12) 

746 

Examples. —  i.  If  a  iio-volt  incandescent  lamp  takes  a  current  of 
.5  ampere,  the  power  required  to  light  it  is  i  lo  x  .5,  or  55  watts.  A  little 
more  than  one  horse  power  (770  watts)  would  be  required  to  light  fourteen 
such  lamps. 

2.  The  power  expended  in  lighting  an  arc  lamp  when  the  difference  of 
potential  between  the  carbons  is  45  volts,  and  the  current  8  amperes,  is  45  X  8, 
or  360  watts  (a  trifle  less  than  half  a  horse  power). 

PROBLEMS 

1.  What  is  the  power  of  a  battery  that  is  able  to  maintain  a  current  of 
4  amperes  through  a  resistance  of  6  ohms? 

2.  A  current  of  5  amperes  is  passed  for  one  minute  through  a  coil  of 
4  ohms'  resistance  in  a  calorimeter  containing  200  g.  of  water.  Neglecting 
the  heat  absorbed  by  the  calorimeter,  what  is  the  rise  of  temperature  of  the 
water? 

3.  A  i6-candle-power  lamp  has  a  resistance  of  200  ohms  when  hot,  and  is 
used  on  iio-volt  circuit.  WTiat  is  the  cost  of  running  the  lamp  at  the  rate  of 
ten  cents  per  kilowatt-hour?  (A  kilowatt-hour  is  a  power  of  one  kilowatt 
supplied  for  one  hour.) 

4.  An  arc  lamp  takes  a  current  of  7  amperes  with  an  E.  M.  T.  of  50  volts. 
What  is  the  cost  of  running  the  lamp  at  eight  cents  per  kilowatt-hour? 


Induced  Currents 


389 


VI.   Induced  Currents 


310. 


492.  Induction  of  Currents  by  Magnets.  —  Electric  currents  can 
be  generated  without  chemical  action  by  means  of  a  process  called 
eleciro-7nagnetic  induction,  or  simply  induction.  Induced  currents 
are  generated  on  a  large  scale 
by  means  of  dynamos.  The 
laws  of  electro-magnetic  induc- 
tion can  be  studied  by  means 
of  the  apparatus  shown  in  Figs. 
310  and  312.  A  coil  of  insu- 
lated wire  is  connected  with  a 
sensitive  galvanometer  (astatic 
or  d'Arsonval) .  While  a  strong 
magnet  is  being  rapidly  inserted 
into  the  hollow  of  the  coil  or 
rapidly  withdrawn  from  it,  the 
galvanometer  indicates  a  mo- 
mentary current ;  but  there  is  no  current  while  the  magnet  remains 
at  rest  within  the  coil.  Inserting  the  north  pole  of  the  mag- 
net causes  a  deflection  of  the  needle  in  one  direction ;  insert- 
ing the  south  pole,  a  deflection  in  the  opposite  direction.  When 
either  pole  is  withdrawn,  the  direction  of  the  current  is  opposite 
to  that  produced  by  inserting  the  same  pole. 

The  induced  current  is  called  direct  if  its  direction  round  the 
coil  is  such  that  like  poles  of  the  coil  and  the  magnet  point  in  the 
same  direction  ;  inverse^  if  their  like  poles  point  in  opposite  direc- 
tions. The  direction  of  the  current  round  the  coil  can  be  deter- 
mined from  its  connection  with  the  galvanometer  and  the  direction 
in  which  the  needle  is  deflected.  It  will  be  found  that  the  current 
is  inverse  when  either  pole  of  the  magnet  is  inserted  and  direct  when 
it  is  luithdrawn  ( Fig.  311). 

Laboratory  Exercise  ^4. 

493.  Source  of  the  Energy  of  the  Induced  Current.  —  It  will  be 
seen  from  Fig.  311  that,  whether  the  magnet  is  being  inserted  or 


390 


Electricity 


Fia  311. 


removed,  its  motion  is  opposed  by  the  magnetic  field  of  the  induced 
current.  Thus  on  account  of  the  induced  current  more  work  must 
be  done  in  moving  the  magnet  either 
into  or  out  of  the  coil  than  would  other- 
wise be  required.  This  additional  work 
is  the  source  of  the  energy  of  the  induced 
current.  The  transformation  of  mechan- 
ical energy  into  the  electrical  energy  of 
the  induced  current  is  supposed  to  be 
effected  through  the  medium  of  the 
ether.  It  takes  place  on  a  large  scale  in 
the  generation  of  currents  by  dynamos. 
494.  Induction  of  Currents  by  Currents.  — The  same  effects  are 
produced  when  a  second  coil,  in  which  a  strong  current  is  flow- 
ing, is  used  instead  of  the  magnet  in  the  preceding  experiments 
(Fig.  312).  The  coil  that  takes  the  place  of  the  magnet  is  called 
the  primaiy  coil^  the 
other  the  secondary 
eoil.  The  primary 
coil  is  connected 
with  a  battery,  and 
its  inductive  action, 
like  that  of  a  mag- 
net, is  due  to  its 
magnetic  field.  The 
inverse  induced  cur- 
rent, which  is  caused 
by  inserting  the  pri- 
mary coil  into  the 
secondary,  is  oppo- 
site in  direction  to 
the  primary  current ;  the  direct  induced  current,  which  is  caused 
by  withdrawing  the  primary  coil,  is  in  the  same  direction  as  the 
primary  current. 

If  the  primary  circuit  is  closed  and  broken  while  the  primary 


Fig.  312. 


Induced   Currents  391 

coil  remains  at  rest  within  the  secondary,  the  effects  are  respec- 
tively the  same  as  when  the  primary  coil  is  inserted  and  with- 
drawn with  the  circuit  closed.  In  either  case  the  induced  current 
is  due  to  the  change  in  the  magnetic  field  within  the  secondary 
coil. 

The  induced  currents  are  in  all  cases  much  stronger  when  the 
primary  coil  contains  a  soft  iron  core.  This  is  because  the  iron 
greatly  increases  the  strength  of  the  magnetic  field. 

495.  Laws  of  Electro- magnetic  Induction.  —  The  following  laws 
of  electro-magnetic  induction  have  been  established  by  experiment. 
The  pupil  should  consider  to  what  extent  the^  are  illustrated  by 
the  experiments  previously  discussed  and  by  subsequent  experi- 
ments with  induction  coils. 

I.  An  increase  in  the  strength  of  the  magnetic  field  within  a 
closed  coil  induces  an  inverse  current^  and  a  decrease  in  the  strength 
of  the  field  induces  a  direct  current. 

It  will  be  helpful  to  remember  that  the  direction  of  an  inverse 
induced  current  is  anti-clockwise  round  the  coil  to  an  observer 
looking  in  the  direction  of  the  lines  of  force  of  the  magnetic 
field  that  causes  the  induction,  and  that  the  direction  of  a  direct 
induced  current  is  clockwise,  when  viewed  in  the  same  manner 

(Fig.  311)- 

II.  The  induced  E.M.F.  (i.e.  the  E.M.F.  of  the  induced 
current^  is  proportional  to  the  rate  of  increase  or  decrease  of  the 
magnetic  field  within  the  coil,  and  also  to  the  number  of  turns  in 
the  coil. 

The  effect  of  the  rate  of  change  of  the  magnetic  field  is  readily 
shown  by  inserting  and  removing  the  primary  coil  at  different 
rates  of  speed  with  an  iron  core  inserted.  The  deflection  is  con- 
siderable when  the  motion  is  rapid,  but  may  be  indefinitely 
decreased  by  moving  the  coil  more  and  more  slowly  {Exp.). 

III.  The  ratio  of  the  induced  E.  M.  F.  to  the  E.  M.  F.  of  the 
current  in  the  primary  coil  is  very  nearly  equal  to  the  ratio  of  the 
number  of  turns  in  the  secondary  coil  to  the  number  of  turns  in 
the  primary. 


392  Electricity 

Thus  if  there  are  150  turns  in  the  primary  coil  and  30,000  in 
the  secondary,  the  E.  M.  F.  of  the  induced  current  is  approxi- 
mately 200  times  as  great  as  that  of  the  primary  current.  This 
principle  is  applied  in  the  generation  of  high  potential  currents 
by  means  of  induction  coils  (Art.  498).  An  induced  current  of 
lower  potential  than  the  primary  is  obtained  when  the  number 
of  turns  in  the  secondary  coil  is  less  than  that  in  the  primary. 
This  principle  is  utilized  in  the  transformer  (Art.  507). 

IV.  There  is  no  gain  of  energy  in  electro-magnetic  induction. 
The  energy  of  the  induced  current  is  derived  either  from 
mechanical  work, 'as  in  the  experiments  with  the  coil  and  the 
magnet  and  in  the  generation  of  currents  by  dynamos,  or  from 
a  current  in  another  circuit  (without  transference  of  electricity 
between  the  circuits),  as  in  making  and  breaking  the  circuit 
through  the  primary  coil  when  at  rest  in  the  secondary  and  in 
the  action  of  induction  coils  and  transformers. 

496.  Historical.  —  Induced  currents  were  discovered  in  1831 
by  Michael  Faraday,  one  of  the  greatest  of  English  physicists. 
Knowing  that  electric  currents  act- upon  magnets,  he  conducted 
a  series  of  experiments  extending  over  seven  years  in  the  attempt 
to  discover  any  action  of  magnets  upon  currents,  and  was  at  last 
rewarded  by  the  discovery  of  electro-magnetic  induction.  Tyndall 
pronounces  this  the  greatest  experimental  result  ever  obtained. 
Its  importance  can  hardly  be  overestimated,  since  the  action  of 
the  dynamo  depends  upon  induction,  and  the  currents  for  all 
industrial  applications  of  electricity  on  a  large  scale  are  generated 
by  dynamos. 

497.  Self-induction.  —  Whenever  a  current  commences  or  ceases 
in  a  coil,  the  current  in  each  turn  exerts  an  inductive  action  upon 
all  the  turns  of  the  coil.  This  action  of  a  current  upon  itself  is 
called  self-induction^  and  the  current  due  to  it  is  often  called  the 
extra  current.  Self-induction  is  very  considerable  if  the  coil  has 
many  turns,  especially  if  it  contains  an  iron  core.  When  the  cur- 
rent is  turned  on  in  a  coil,  the  growth  of  its  own  magnetic  field 
has  the  same  effect  as  if  a  magnet  were  suddenly  thrust  into  the 


Induced  Currents 


393 


coil.  The  inverse  E.  M.  F.  thus  induced  opposes  the  current, 
preventing  an  immediate  rise  to  its  full  value.  When  the  circuit 
is  broken,  the  effect  of  self-induction  is  the  same  as  if  a  magnet 
were  suddenly  withdrawn  from  the  coil.  The  induced  E.  M.  F. 
in  this  case  is  direct.  It  is  generally  many  times  greater  than 
the  original  or  pri- 
mary E.  M.  F.,  and 
continues  the  cur- 
rent across  the  air 
gap  where  the  cir- 
cuit is  broken,  caus- 
ing a  spark. 

This  effect  of  self- 
induction  can  be 
shown  as  follows : 
A  file  is  connected 
to  one  pole  of  a 
chromic  acid  bat- 
tery and  a  piece  of  wire  to  the  other.  As  the  free  end  of  this 
wire  is  drawn  over  the  file,  the  circuit  is  rapidly  closed  and  broken, 
and  at  each  break  a  minute  spark  occurs.  When  the  experiment 
is  repeated  with  the  coils  of  a  large  electro-magnet  included  in  the 
circuit  (Fig.  313),  brilliant  sparks  are  obtained,  indicating  a  high 
induced  E.  M.  F.  at  each  break  {Exp.). 

498.  The  Induction  Coil.  —  The  indue fion  or  Ruhmkorff  coil 
(Fig.  314)  is  an  instrument  for  generating  induced  currents  of 
very  high  potential.  A  simplified  diagram  of  the  essential  parts 
is  shown  in  Fig.  315.  These  are  an  iron  core,  AB  \  a  primary 
coil  consisting  of  one  or  two  layers  of  turns  of  large  insulated  wire  ; 
a  secondary  coil  of  very  fine  wire,  well  insulated  and  often  many 
miles  in  length  ;  an  automatic  make-and-break  device,  or  current 
interrupter,  CD,  which  is  included  in  the  primary  circuit ;  and  a 
condenser,  E.  There  is  generally  also  a  device,  called  a  confinu- 
tator,  for  reversing  the  current  through  the  primary  coil  without 
changing  the  battery  connections ;  but  this  is  not  essential. 


Fig.  313. 


394 


Electricity 


When  a  current  from  a  battery  is  sent  through  the  primary  coil, 
it  magnetizes  the  iron  core,  and  the  core  attracts  the  iron  block 


I'iG.  314. 

C,  which  IS  supported  near  the  end  of  the  core  upon  a  spring. 
This  spring  is  the  movable  part  of  the  interrupter,  and  the  pri- 
mary current  passes  between  it  and  the  point  of  a  screw,  D,  against 
which  it  rests.  By  the  attraction  of  the  magnetized  core  the 
spring  is  drawn  away  from  the 
point,  breaking  the  circuit.  The 
core  instantly  becomes  demag- 
netized, and  the  spring  flies  back 
again,  closing  the  circuit.  The 
primary  circuit  is  thus  closed  and 
broken  many  times  every  second, 
causing  alternately  an  inverse 
and  a  direct  induced  E.  M.  F.  in 
the  secondary  coil.     The  ends  ^'^-  3^5- 

of  the  secondary  coil  are  connected  with  the  binding  posts  R  and 
G^  and  may  be  extended  by  means  of  rods  or  wires  attached  to 
the  posts  until  the  gap  H  is  made  as  small  as  desired.  When 
this  gap  is  not  too  great,  an  electric  spark  passes  between  the 
terminals  with  every  interruption  of  the  primary  current.    The 


Induced  Currents  395 

length  of  the  spark  that  can  be  obtained  increases  with  the  in- 
duced E.  M.  F.,  and  varies  in  different  coils  from  a  few  milli- 
meters to  30  cm.  or  more  for  very  powerful  coils.  It  is  estimated 
that  an  E.  M.  F.  of  30,000  volts  is  required  to  cause  a  spark 
across  a  distance  of  i  cm.  in  air  under  atmospheric  pressure  and 
at  ordinary  temperatures ;  and  that,  under  the  same  conditions, 
an  E.  M.  F.  of  at  least  300  or  400  volts  is  required  to  start  a  spark, 
however  short. 

The  E.  M.  F.  of  the  induced  current  increases  with  the  number 
of  turns  in  the  secondary  coil  and  with  the  rate  of  change  of  the 
magnetic  field  (Art.  495,  Law  II).  The  latter  depends  upon  the 
rate  of  increase  or  decrease  of  the  current  in  the  primary  coil,  both 
of  which  are  retarded  by  self-induction.  The  stopping  of  the 
primary  current  is,  however,  much  more  abrupt  than  the  starting 
and  induces  a  correspondingly  greater  E.  M.  F.  In  general,  it  is 
only  at  the  "  break  "  of  the  primary  circuit  that  a  spark  occurs 
between  the  terminals  of  the  secondary  coil.  The  purpose  of  the 
condenser  is  to  prevent  or  at  least  diminish  the  spark  at  the  inter- 
rupter, and  thus  increase  the  abruptness  of  the  "  break."  This  it 
does  by  serving  as  a  temporary  reservoir  into  which  the  extra  cur- 
rent flows  instead  of  jumping  across  the  gap.  (The  construction 
and  action  of  a  condenser  are  discussed  in  Art.  524.) 

499.  Effects  of  the  Induced  Current.  —  While  the  E.  M.  F.  of 
the  induced  current  is  enormously  higher  than  that  of  the  bat- 
tery current  through  the  primary  coil,  its  strength  is  exceedingly 
small.  The  induced  current  may  be  compared  to  water  flowing 
through  a  very  small  pipe  under  very  great  pressure,  and  the  pri- 
mary current  to  water  flowing  through  a  large  pipe  under  very  little 
pressure. 

Although  the  energy  of  the  induced  current  is  necessarily  less 
than  the  total  energy  of  the  primary  current,  the  induced  current 
is  nevertheless  capable  of  producing  effects  that  are  impossible 
with  the  primary  current.  For  example,  a  spark  3  or  4  cm.  long 
is  capable  of  piercing  a  sheet  of  cardboard  or  thin  pieces  of  other 
nonconductors  {Exp.).    The  induced  current  is  also  capable  of 


396 


Electricity 


producing  a  shock  and  other  physiological  effects.  These  are  the 
well-known  effects  of  the  physician's  battery,  which  is  a  small 
induction  coil  operated  by  a  voltaic  cell.  The  handles  which  are 
held  by  the  patient  are  the  terminals  of  the  secondary  coil.  The 
current  from  a  powerful  induction  coil  is  very  painful  and  may  even 
be  dangerous. 

The  induction  coil  is  also  used   for   obtaining   discharges   in 
rarefied  gases  inclosed  in  glass  tubes  and  bulbs  (Art.  526). 


VII.  The  Dynamo  and  the  Motor 

500.  The  Principle  of  the  Dynamo.  —  Suppose  a  single  loop  of 
wire  to  be  mounted  on  an  axis  at  right  angles  to  the  lines  of  force 
of  a  strong  magnetic  field  as  shown    in    Fig    316.     When    the 

loop  is  vertical,  the  portion  of  the 
magnetic  field  extending  through  it 
in  the  direction  of  the  lines  of  force 
is  as  great  as  possible.  As  the  loop 
is  turned  from  li.is  position,  the  cross- 
section  of  the  field  extending  through 
it  (always  taken  at  right  angles  to 
Fig.  316.  jj^g  Ijj^gg  Qf  force)  decreases,  and  be- 

comes zero  when  the  loop  is  horizontal.  When  the  magnetic 
field  is  thus  Removed  from  within  the  loop,  the  inductive  action 
is  the  same  as  if  it  were  removed  in  any  other  way.  With 
the  direction  of  rotation  indicated  by  the  arrow,  the  induced 
current  is  clockwise,  looking  in  the  direction  of  the  lines  of 
force.  As  the  rotation  is  continued  in  the  same  direction  from 
the  horizontal  to  the  vertical  position  again,  an  increasing  cross- 
section  of  the  magnetic  field  extends  through  the  coil.  This 
induces  an  anti-clockwise  current,  looking  in  the  same  direction  as 
before  ;  but,  as  the  opposite  face  of  the  loop  is  now  turned  toward 
the  observer,  the  direction  of  the  current  round  the  loop  is  the 
same  as  before.  As  the  loop  passes  the  vertical,  the  current  is 
reversed  in  it,  and  continues  thus  till  it  passes  through  the  vertical 


The  Dynamo  and  the  Motor  397 

again.  (Why?)  Thus  a  continuous  rotation  of  the  loop  induces 
an  alternating  current  in  it,  the  reversal  of  the  current  taking  place 
twice  during  each  rotation,  as  the  loop  passes  through  the  position 
at  right  angles  to  the  lines  of  force. 

With  a  constant  rate  of  rotation,  the  cross-section  of  the  mag- 
netic field  extending  through  the  loop  changes  most  rapidly  when 
the  loop  is  in  .the  neighborhood  of  the  horizontal  position,  and 
least  rapidly  in  the  neighborhood  of  the  vertical  position ;  hence 
the  induced  E.  M.  F.  is  greatest  in  the  horizontal  position  and  least 
in  the  vertical.  In  fact,  it  diminishes  to  zero  in  the  latter  position 
at  the  instant  it  reverses  in  direction.  The  induced  current  passes 
through  like  changes.^ 

The  current  thus  generated  in  the  loop  can  be  sent  through  an 
external  circuit  by  means  of  either  device  shown  in  Figs.  317 
and  318.     In  the  first  case  each  end  of  the  loop  is  connected 


Fig.  317. 


Fig.  318. 


with  a  copper  ring  surrounding  the  axis.  The  terminals  of  the 
external  circuit  are  connected  with  copper  strips,  called  brushes^ 
which  press  against  the  rings  and  make  sliding  contact  with  them 
as  they  rotate.  The  current  in  the  external  circuit  reverses  with 
every  reversal  in  the  loop.  This  is  the  principle  of  the  alternating- 
current  dynamo.  In  the  second  case  the  ends  of  the  loop  are 
connected  with  the  two  halves  of  a  split  copper  tube.  This  device 
is  called  a  commutator.  The  brushes  are  adjusted  so  that  the 
contact  of  each  changes  from  one  segment  of   the  commutator 

1  If  the  lines  of  force  of  the  field  were  vertical,  it  would  be  necessary  to  read  hori- 
zontal for  vertical,  and  v/V<f  versa,  in  the  above  discussion  ;  for  the  inductive  action 
depends  upon  the  relative  positions  and  motion  of  the  loop  and  the  lines  of  force. 


39^  Electricity 

to  the  other  when  the  current  reverses  in  the  loop;  hence  the 
current  in  the  external  circuit  is  direct,  i.e.  it  always  flows  in  the 
same  direction.  This  is  the  principle  of  the  direct-current  dynamo. 
501.  Dynamo-electric  Machines.  —  The  E.  M.  F.  induced  in  a 
coil  by  rotating  it  in  a  magnetic  Jield  is  proportional  ( i )  to  the 
number  of  turns  in  the  coil,  (2)  to  the  strength  of  the  magnetic 
field,  and  (3)  to  the  rate  of  rotation.  It  is  by  the  application  of 
these  principles  that  electrical  power  is  developed  on  a  large  scale 
by  means  of  dynamos.  The  current  is  generated  in  a  series  of 
coils  of  wire,  called  the  armature,  which,  in  the  more  common 
form  of  dynamo,  is  rotated  at  high  speed  between  the  poles  of  a 
powerful  electro- magnet,  called  the  field  magnet.  In  some  large 
dynamos  the  field  magnet  has  several  poles ;  and  powerful  alter- 
nating-current dynamos  are  sometimes  made  with  stationary 
armatures  and  revolving  field  magnets.  In  all  cases  the  funda- 
mental principle  is  the  same,  —  an  alternating  current  is  gener- 
ated in  the  coils  of  the  armature  by  a  rapid,  periodic  change  of 
the  magnetic  field  within  them. 


Fig.  319. 

502.   The  Direct-current  Dynamo.  —  The  Armature. — The  coils 
of  an  armature  are  wound  upon  a  core  of  soft  iron,  which  greatly 


The  Dynamo  and  the   Motor  399 

intensifies  the  magnetic  field  within  them.  This  core  is  generally 
in  the  form  of  a  ring  or  a  long  cylinder.  An  armature  with  a  ring 
core  is  called  a  ring  armatiwe ;  one  with  a  cylindrical  core  is 
called  a  drum  armature.  The  dynamo  shown  in  Fig.  319  has  a 
drum  armature.  The  wire  is  wound  in  a  number  of  coils,  which 
extend  lengthwise  and  across  the 
ends  of  the  core.  The  plan  of  a 
four-coil  armature  is  shown  in  Fig. 
320.  In  practice  the  number  of 
coils  is  much  greater  than  this.  The 
coils  are  connected  in  series  and  lie 
in  different  positions  about  the  axis 
of  the  core ;  as  a  result  of  which  the 
induced  E.  M.  F.  is  practically  con- 
stant. The  ends  of  the  coils  are  also  yw,.  320. 
connected  to  the   segments  of  the 

commutator  as  shown  in  the  figure.  As  the  segments  are  insu- 
lated from  each  other,  the  current  generated  in  any  coil  can  reach 
the  positive  brush  only  by  passing  in  series  through  all  the  coils 
between  it  and  the  one  whose  commutator  segment  is  at  that 
instant  in  contact  with  the  brush. 

The  Field  Magnet.  —  While  a  dynamo  is  in  operation,  the  field 
magnet  must  be  strongly  magnetized.  This  is  accomplished  in  a 
direct-current  dynamo  by  sending  a  part  or  all  of  the  current 
generated  by  it  through  the  coils  of  the  field  magnet.  In  either 
case  the  energy  consumed  in  the  coils  of  the  magnet  is  only  a 
very  small  fraction  of  the  whole.  In  starting  a  dynamo,  the 
current  is  at  first  very  weak,  since  the  field  magnet  retains  only 
slight  traces  of  magnetization;  but  this  small  current  flowing 
through  the  coils  of  the  magnet  strengthens  it,  causing  a  stronger 
current.  This  mutual  action  continues  until  the  magnet  becomes 
fully  magnetized. 

503.  Transformation  of  Energy  in  a  Dynamo.  —  We  have  seen 
that,  when  a  coil  in  which  a  current  is  flowing  is  placed  in  a 
magnetic   field,  the    magnetic  forces  tend  to  turn  it  so  that  its 


400 


Electricity 


north  end  (or  side)  will  face  in  the  direction  of  the  lines  of  force 
of  the  field,  as  a  magnet  would  do  (Arts.  456  and  484).  It 
follows  from  this,  as  the  pupil  may  prove  with  the  aid  of  Fig.  316, 
that,  as  a  coil  is  rotated  in  a  magnetic  field,  the  magnetic  forces 
acting  upon  the  coil  in  consequence  of  the  induced  current  flow- 
ing in  it  tend  to  turn  it  in  the  opposite  direction.  It  can  be 
shown  that  the  magnetization  induced  in  the  core  of  the  arma- 
ture by  the  current  also  opix)ses  the  rotation  by  which  the  current 
is  induced. 

As  a  result  of  these  opposing  magnetic  forces,  a  much  greater 
expenditure  of  energy  is  required  to  rotate  an  armature  while  a 
current  is  being  generated  than  is  necessary  when  the  circuit  is 
open.  An  armature  that  requires  10  horse  power  to  run  it  while 
generating  a  current  can  be  run  at  equal  speed  on  open  circuit  by 
a  fraction  of  a  horse  power.'  The  additional  energy  delivered 
to  the  dynamo  when  the  circuit  is  closed  is  transformed  by  the 
opposing  electro- magnetic  forces  into  the  energy  of  the  induced 
current. 

504.   The  Direct-current  Motor.  —  When  a  current  from  some 

other  source  is  sent  through  the  coils 
of  the  field  magnet  and  the  arma- 
ture of  a  dynamo  in  the  same  direc- 
tion as  the  current  that  the  dynamo 
itself  would  generate,  the  magnetic 
forces  acting  upon  the  armature 
cause  it  to  turn  in  the  opposite 
direction  to  that  which  would  gen- 
erate the  current.  The  machine 
then  runs  as  a  motor.  The  rotation 
is  in  the  same  direction  both  as 
a  motor  and  as  a  dynamo  if  the 
current  is  sent  through  the  coils 
Fig.  321.  of  the   field-magnet  in  the  same 

1  The  difference  in  the  power  required  on  ctosed  and  on  open  circuit  is  very 
noticeable,  even  with  a  small  hand-power  dynamo  (Exp.). 


The  Dynamo  and  the   Motor  401 

direction  in  both  cases  and  through  the  armature  in  opposite 
directions.  Any  direct-current  dynamo  will  run  as  a  motor ; 
but,  for  practical  reasons,  motors  are  made  slightly  different  from 
dynamos. 

Figure  321  represents  a  simplified  diagram  of  a  small  motor  hav- 
ing a  three-coil  armature.  The  current  divides  at  the  positive 
brush,  part  going  through  one  coil  of  the  armature  in  one  direc- 
tion and  part  through  the  other  two  coils  in  the  opposite  direction. 
This  makes  one  outer  pole  of  the  armature  north  and  the  other 
two  south,  or  vice  versa,  according  to  the  position  of  the  armature. 
By  change  of  contact  between  the  commutator  segments  and  the 
brushes,  the  current  reverses  in  each  coil  as  it  passes  through  the 
horizontal  position  on  either  side,  the  direction  of  the  current  in 
the  coils  being  always  such  that  the  attractions  and  repulsions 
between  the  poles  of  the  field-magnet  and  the  poles  of  the  arma- 
ture all  act  in  the  same  direction  round  the  axis,  causing  rotation. 

Laboratory  Exercise  75. 

505.  Transformation  of  Energy  in  the  Motor.  —  When  the 
current  from  a  battery  is  sent  through  a  small  motor,  a  galva- 
nometer (^,  Fig.  322)  placed  in  the  circuit  will  show  that  the  cur- 
rent is  much  smaller  when  the  motor  is  running  than  it  is  when 
the  armature  is  held  stationary.  By  permit- 
ting the  motor  to  run  at  different  speeds,  it 
will  be  found  that  the  current  decreases  as 
the  speed  increases.  At  the  same  time  a 
voltmeter  (K  in  the  figure)  will  show  that 
the  fall  of  potential  in  the  armature  increases 
as  the  speed  increases  {Exp.). 

The  fall  of  potential  in  the  armature  when  '^"  ^"' 

at  rest  is  no  more  than  is  necessary  to  send  the  given  current 
through  the  resistance  of  the  coils  (Art.  475).  When  the  armature 
is  rotating,  an  additional  fall  of  potential  is  required  to  overcome 
an  opposing  E.  M.F.  induced  in  the  armature  in  consequence  of 
its  rotation ;  for  the  machine,  while  running  as  a  motor,  tends  as 
a  dynamo  to  generate  a  current  in  the  opposite  direction.     This 


402 


Electricity 


counter  E.  M.  F.  increases  with  the  speed,  and  is  the  cause  of  the 
decrease  of  the  current. 

If  the  armature  of  a  motor  is  not  permitted  to  turn,  the  energy 
lost  by  the  current  in  the  motor  is  all  converted  into  heat  in 
overcoming  the  resistance  of  the  coils,  as  in  any  other  conductor 
(Art.  490)  ;  but  when  a  motor  is  running,  part  of  the  electrical 
energy — generally  80  to  90  per  cent  of  it  —  is  expended  in  over- 
coming the  counter  E.  M.  F.  in  the  armature.  It  is  this  energy 
that  is  transformed  into  mechanical  energy  by  the  motor  in  doing 
work;  and  the  rate  at  which  the  work  is  done  is  measured  in 
watts  by  the  product  of  the  current  and  the  counter  E.  M.  F. 

506.  Transmission  of  Electrical  Energy.  —  In  transmitting  elec- 
trical energy  over  a  line  for  use  at  a  distance,  a  portion  of  it  is 
lost  as  heat.  This  loss  is  due  to  the  resistance  of  the  conductor, 
and  limits  the  distance  to  which  it  is  practicable  to  transmit  power 
by  electricity.  The  conditions  affecting  the  amount  of  power  that 
is  lost  in  transmission  are  as  follows :  Let  E^  denote  the  E.  M.  F. 
of  the  current  at  the  start  and  E^  its  E.  M.  F.  at  the  farther  end  of 
the  line,  C  the  strength  of  the  current,  and  R  the  resistance  of  the 
line.  The  power  generated  is  ^jC  watts,  and  the  power  delivered 
for  use  E2C  watts  (Art.  491).  The  fall  of  potential  in  the  line  is 
Ex — Ei  volts,  and  the  power  lost  in  transmission  {E^ — E^)  C  or 
C'R  watts.  Numerical  results  obtained  from  these  formulas  with 
assumed  values  for  E^t  C,  and  R  are  given  for  comparison  in  the 
following  table :  — 


Given :  — 

z 

3 

3 

4 

E.  M.  F.  delivered  {E^, 

500  volts 

500  volts 

1000  volts 

5000  volts 

Current  (C), 

10  amperes 

20  amperes 

10  amperes 

2  amperes 

Resistance  of  line  {/d). 

.25  ohms 

25  ohms 

25  ohms 

25  ohms 

Then  : — 

Power  delivered  i^E^C), 

5000  watts 

xo,ooo  watts 

10,000  watts 

10,000  watts 

Power  lost  {C^R), 

2500  watts 

10,000  watts 

2500  watts 

100  watts 

Power  generated  {E^C), 

7500  watts 

20,000  watts 

12,500  watts 

10,100  watts 

Fraction  of  power  lost, 

33-3% 

50% 

20% 

1%  (nearly) 

E.M.F.  at  the  start  (£■!), 

750  volts 

1000  volts 

1250  volts 

5050  volts 

The  Dynamo  and  the  Motor  403 


It  will  be  seen  from  the  formulas  and  from  the  numerical 
examples  that  the  power  delivered  over  a  given  line  can  be 
increased  by  increasing  either  the  current  or  the  voltage  at  which 
it  is  delivered,  but  that  the  former  greatly  increases  the  relative 
amount  of  power  lost,  while  the  latter  decreases  it.  Hence  the 
economical  transmission  of  electrical  energy  over  long  distances 
requires  currents  of  high  potential. 

In  recent  years  electrical  power  stations  have  been  established 
on  a  large  scale  at  Niagara  Falls,  at  various  points  in  the  Sierra,, 
Nevada  Mountains,  and  elsewhere,  where  water  power  is  utilized  in 
running  dynamos  of  3000  to  5000  horse  power.  The  currents 
generated  at  these  stations  are  transmitted  at  an  electrical  pressure 
of  40,000  to  60,000  volts  to  surrounding  cities,  in  some  cases  to  a 
distance  of  200  miles.  Power  thus  obtained  is  rapidly  taking  the 
place  of  the  steam  engine  in  manufacturing,  mining,  and  trans- 
portation. 

507.  The  Transformer.  —  The  high-potential  currents  that  are 
used  for  transmitting  electrical  power  over  long  distances  are  not 
adapted  to  any  of  the  ordinary  uses  of  electrical  power,  and, 
besides,  are  extremely  dangerous.  The  potential  must  therefore 
be  reduced  without  unnecessary  loss  of  power  at  or  near  the  place 


Alternator 


Loir  Pressure  ^fains 
lAimps 


Higli  Pressure  Mains 


Fig.  323. 


Lamps 


where  the  power  is  to  be  used.  With  alternating  currents  this 
can  be  accomplished  by  means  of  the  transformer  (Fig.  323). 
This  instrument  is  merely  a  reversed  induction  coil,  consisting  of 
a  primary  coil  of  many  turns  and  a  secondary  coil  of  fewer  turns, 
wound  on  the  same  core  of  soft  iron.  The  high-potential  current 
is  sent  through  the  primary  coil,  and,  by  its  rapid  reversals  (from 


404  Electricity 

50  to  120  per  second),  induces  an  alternating  current  in  the  sec- 
ondary coil.  If  there  are  -  as  many  turns  in  the  secondary  coil  as 
there  are  in  the  primary,  the  E.  M.  F.  of  the  induced  current  will 

be  approximately  -  as  great  as  that  of  the  primary,  and  its  strength 
n 

n  times  as  great  (Art.  495,  Law  III).  The  loss  of  power  in  the 
transformation  is  small.  The  motors  or  lamps  to  be  supplied  are 
connected  in  the  circuit  of  the  secondary  coil. 

Alternating-current  dynamos  and  motors  are  not  discussed  in 
this  book.  They  differ  in  some  important  respects  from  direct- 
current  machines. 

PROBLEMS 

1.  Could  the  current  generatefl  l)y  a  dynamo  be  used  to  run  a  motor  while 
the  motor  was  running  the  dynamo? 

2.  What  relation  to  the  law  of  the  conservation  of  energy  has  the  fact 
that  more  power  is  requireil  to  run  a  dynamo  on  closed  circuit  than  on  open 
circuit? 

3.  A  direct-current  dynamo  delivers  a  current  of  50  amperes  at  a  pressure 
of  500  volts  to  a  line  whose  resistance  is  2  ohms.  Find  (//)  the  power  gener- 
ated, (*)  the  power  lost  in  the  line,  and  (<•)  the  power  delivered.  (</)  What 
per  cent  of  the  power  is  lost  in  the  line? 

4.  What  per  cent  of  the  power  would  be  lost  in  the  line  if  the  dynamo 
generated  25  amperes  at  a  pressure  of  looo  volts? 

VIII.    The  Telephone  and  the  Microphone 

508.  The  Telephone. — The  instrument  known  as  the  receiver 
of  a  telephone  (Fig.  324)  was  invented  by  Graham  Bell  in  1876, 
and  was  used  by  him  both  as  a  receiver  and  as  a  transmitter.  It 
contains  a  permanent  bar  magnet,  A^  one  end  of  which  is  sur- 
rounded by  a  coil,  B^  of  fine  copper  wire.  This  coil  is  connected 
with  the  binding  posts  at  the  end  of  the  handle.  A  disk  of  very 
thin  sheet  iron,  EE,  extends  across  the  receiver  at  a  distance  of 
about  a  millimeter  from  the  end  of  the  magnet.  This  disk  is 
forced  into  vibration  by  the  sound  waves  that  beat  upon  it  when 
the  instrument  is  held  before  the  mouth  in  speaking.      The  mag- 


The  Telephone  and  the  Microphone     405 


Fio.  324. 


net  is  strengthened  as  the  disk  moves  toward  it  in  vibrating,  and 
weakened  as  the  disk  recedes.  This  induces  alternating  currents 
in  the  coil,  as  in  the  armature 
of  a  dynamo. 

The  current  thus  gener- 
ated flows  through  the  coil 
of  tlie  receiver  at  the  other 
end  of  the  line,  alternately 
increasing  and  decreasing 
the  strength  of  the  magnet  as  it  flows  first  in  one  direction,  then 
in  the  other  round  the  coil.  When  the  magnet  is  strengthened, 
it  draws  the  disk  more  strongly;  when  it  is  weakened,  the  disk 
springs  back.  The  disk  of  the  receiving  telephone  thus  re- 
l)eats  the  movements  imparted  by  the  sound  waves  to  the  disk 
of  the  transmitting  telephone ;  and,  by  its  vibration,  reproduces 
tlie  sound  in  pitch,  relative  intensity,  and  quality  with  remarkable 
accuracy. 

A  telephone  line  of  this  sort  does  not  require  a  battery.  It 
works  successfully  over  short  distances ;  but  over  a  long  line  the 
current  is  too  weak  to  reproduce  the  sound  with  sufficient  inten- 
sity. The  Bell  telephone  has  continued  in  use  as  a  receiver, 
but  has  been  superseded  as  a  transmitter  by  a  special  form  of 
microphone. 

509.   The  Microphone.  —  This  instrument,  as  its  name  implies, 

increases  the  intensity  of  faint 
sounds.  A  common  form  of 
microphone  is  shown  in  Fig.  325. 
The  pointed  ends  of  a  stick  of 
carbon  rest  loosely  in  cavities  in 
two  horizontal  supports,  also  of 
carbon.  These  supports  are 
fixed  to  a  small  sounding  board, 
and  are  connected  by  means  of 
binding  posts  with  a  telephone 
receiver  and  a  battery. 


Fig.  325. 


4o6 


Electricity 


Vibrations  of  the  sounding  board,  due  either  to  the  presence  of 
a  sounding  body  or  to  any  slight  disturbance  of  the  instrument, 
produce  rapid  changes  of  pressure  at  the  points  of  contact  of  the 
carbons  with  one  another.  The  resistance  at  these  points  of 
contact  decreases  when  the  pressure  increases,  and  vice  versa^ 
producing  an  unsteady  current  in  the  circuit.  This  causes  the 
telephone  receiver  to  emit  sounds  which  are  similar  to  the  original 
sounds,  but  generally  very  much  louder. 

510.  The  Transmitter.  —  The  Bell  telephone  has  been  super- 
seded as  a  transmitter  by  different  forms  of  microphones.  Figure 
326  represents  the  Blake  transmitter.  Back  of  the 
mouthpiece  there  is  a  thin  disk,  ^/,  which  is  set  in 
vibration  by  the  voice.  The  back  of  the  disk  touches 
a  platinum  point  attached  to  the  end  of  a  light 
spring,  /.  A  block  of  carbon,  k,  at  the  end  of  a 
stiffer  spring,  g,  also  presses  lightly  against  the  plati- 
num point,  completing  a  local  battery  circuit  through 
the  springs  and  the  primary  of  a  small  induction 
coil  (Fig.  327).  The  receiver  and  the  secondary  of 
the  induction  coil  are  included  in  the  line  circuit 
which  connects  with  the  central  station. 

The  vibration  of  the  disk  causes  alternate  increase  and  decrease 
of  pressure  and  corresponding  changes  of  resistance  at  the  point 
of  contact  of  the  platinum  and 
the  carbon.  The  changes  thus 
produced  in  the  battery  current 
cause  induced  currents  in  the 
secondary  of  the  induction  coil, 
and  these  induced  currents  cause 
the  receiver  at  the  other  end  of 
the  line  to  reproduce  the  message. 


OtEarik 


Fk;.  327. 

The  lever  upon  which  the  receiver  of  a  telephone  is  hung  when 
not  in  use  is  pulled  down  by  the  weight  of  the  receiver.  When 
the  lever  is  in  this  position,  the  battery  circuit  is  broken  and  the 
call-bell  is  connected  with  the  line.     When  the  receiver  is  taken 


Chemical  Effects  of  Currents 


4»7 


down,  the  lever  is  raised  by  a  spring.  This  automatically  discon- 
nects the  bell  from  the  line,  connects  the  line  with  the  receiver 
and  the  secondary  of  the  induction  coil,  and  closes  the  battery 
circuit  through  the  transmitter.  The  telephone  is  then  ready  for 
use  either  in  sending  or  receiving  a  message. 
Laboratoiy  Exercise  77. 


IX.   Chemical  Effects  of  Currents* 

511.  Electrolysis.  —  By  the  chemical  actions  that  take  place 
in  a  battery,  chemical  potential  energy  is  liberated  and  converted 
into  electrical  energy.  The  reverse  transformation  is  also  pos- 
sible ;  that  is,  an  electric  current  can  produce  chemical  changes 
by  which  the  energy  of  the  current  is  stored  up  as  chemical  poten- 
tial energy.  For  example,  the  energy  of  a  battery  current  is 
generally  derived  from  the  action  of  sulphuric  acid  on  zinc,  by 
which  zinc  sulphate  is  formed  (Art.  443).  Conversely,  zinc  can 
be  separated  from  zinc  sulphate  by  means 

of  an  electric  current,  as  in  the  following  - — i,^^  r:— 

experiment.  A  bent  tube  (Fig.  328)  is 
partly  filled  with  a  solution  of  zinc  sul- 
phate. A  narrow  strip  of  platinum,  sol- 
dered to  a  wire,  is  placed  in  the  liquid  in 
one  arm  of  the  tube  and  connected  with 
the  positive  pole  of  a  battery  of  two  or 
three  chromic  acid  cells  in  series ;  and 
an  iron  or  a  copper  wire  is  inserted  in 
the  other  arm  of  the  tube  and  connected  with  the  other  pole  of  the 
battery.  While  the  current  is  flowing,  bubbles  continue  to  rise 
from  the  platinum  terminal ;  and,  after  a  few  minutes,  it  will  be 
found  that  the  iron  or  copper  wire  is  covered  with  a  layer  of  zinc. 
The  current  in  passing  through  the  liquid  decomposes  the  zinc 
sulphate  and  carries  the  zinc  with   it   to  the  negative  terminal. 


Fig.  328. 


1  If  the  teacher  wishes  to  discuss  the  theory  of  electrolysis,  this  section  should 
be  preceded  by  the  one  on  electrostatics. 


4o8  Electricity 

The  other  part  of  the  sulphate  (consisting  of  sulphur  and  oxygen) 
goes  in  the  opposite  direction  and  appears  at  the  positive  terminal. 
Here  it  unites  with  hydrogen  from  the  water,  forming  sulphuric 
acid  and  setting  oxygen  free.  This  oxygen  forms  the  bubbles  that 
rise  from  the  strip  of  platinum  {Exp.), 

This  experiment  illustrates  the  decomposition  of  a  chemical 
compound  by  the  passage  of  an  electric  current  through  it.  The 
process  is  called  eUctrolysis,  the  substance  decomposed  is  called 
an  eUctrolyU^  and  the  vessel  in  which  the  process  is  carried  out 
is  called  an  electrolytic  cell.  The  terminal  by  which  the  current 
enters  the  liquid  is  called  the  positive  electrode,  or  anode ;  the  one 
by  which  it  leaves  the  liquid  is  the  negative  electrode,  or  cathode. 

If  an  anode  of  zinc  be  substituted  for  the  platinum  in  the  above 
experiment,  the  compound  of  sulphur  and  oxygen  that  comes  to 
it  wll  unite  with  it,  forming  more  zinc  sulphate.  The  anode  will 
lose  zinc  and  the  cathode  will  gain  it  at  the  same  rate,  and  the 
strength  of  the  solution  will  remain  constant.  Similar  results  are 
obtained  with  copper  electrodes  and  a  solution  of  copper  sulphate. 
The  anode  wastes  away  and  an  equal  weight  of  copper  is  deposited 
upon  the  cathode.  Water  can  be  electrolyzed  in  a  cell  containing 
dilute  sulphuric  acid  and  platinum  electrodes.  The  acid  takes  part 
in  the  chemical  changes  ;  but  it  is  only  the  water  that  is  consumed, 
hydrogen  appearing  at  the  cathode  and  oxygen  at  the  anode. 

512.  Industrial  Applications  of  Electrolysis.  —  "The  earliest 
industrial  application  of  electrolysis  was  in  electrotyping  and  elec- 
troplating. These  operations  consist  in  depositing  a  thin  film  of 
metal  upon  a  surface.  They  are  fundamentally  the  same,  though 
copper  is  the  only  metal  used  for  producing  electrotypes. 

Electrotyping.  —  "  Electrotypes  are  exact  reproductions  of  the 
original  objects.  The  process  of  electrotyping  is  substantially 
as  follows  :  The  page  of  type,  or  the  woodcut,  is  first  reproduced 
in  wax  or  plaster.  This  exact  impression  is  next  covered  with 
powdered  graphite  to  make  it  conduct  electricity.  The  coated 
mold  is  then  suspended  as  the  cathode  in  an  acid  solution  of 
copper  sulphate ;  the  anode  is  a  plate  or  bar  of  copper.     When 


Chemical  Effects  of  Currents 


409 


the  current  is  passed,  electrolysis  occurs ;  copper  is  dissolved  from 
the  anode  and  deposited  upon  the  mold  in  a  film  of  any  desired 
thickness.  The  exact  copper  copy  is  stripped  from  the  mold, 
backed  with  metal,  and  mounted  on  a  wooden  block,  and  used 
instead  of  the  type  or  woodcut  itself.  By  this  process  exact  copies 
of  expensive  wood  engravings  can  be  cheaply  reproduced,  and 
type  can  be  saved  from  the  wear  and  tear  of  printing.  Most  books, 
magazines,  and  newspapers  are  now  printed  from  electrotypes. 

Electroplating.  —  "The  process  of  electroplating  differs  from 
electrotyping  in  only  one  essential ;  viz.,  in  electroplating  the 
deposited  film  is  not  removed  from  the  object.  The  object  to  be 
plated  is  carefully  cleaned  and  made  the  cathode  in  a  solution  of 
some  salt  of  the  metal  to  be  deposited ;  the  anode  is  a  bar  or 


Fig.  329. 

plate  of  the  same  metal  (Fig.  329).  When  the  current  passes 
through  the  system,  the  metal  is  firmly  deposited  upon  the  object. 
The  electrolysis  would  take  place,  of  course,  if  any  anode  were 
present ;  but  anodes  of  the  metal  to  be  deposited  are  usually  used 
to  prevent  the  solution  or  '  bath  '  from  weakening.  They  accom- 
plish the  purpose  by  replenishing  the  solution  with  metal  as  fast 
as  it  is  removed  and  deposited  upon  the  cathode..  Silver,  nickel, 
and  gold  are  the  metals  generally  used  in  electroplating. 

Electro-metallurgy.  —  "  It  is  only  within  the  last  ten  or  fifteen 
years  that  the  electric  current  has  been  profitably  applied  to  many 


4IO 


Electricity 


industries.  But  during  this  time  the  development  of  electro-chem- 
istry has  been  very  marked.  The  largest  of  these  industries  is  the 
refining  of  copper.  The  process  is  similar  to  that  described 
under  electrotyping.  Other  metals,  such  as  gold,  silver,  and  lead, 
are  extracted  from  their  ores  and  purified  by  electricity,  though 
the  older  processes  are  still  used.  All  the  aluminum,  magnesium, 
and  sodium  of  commerce  are  now  manufactured  by  passing 
an  electric  current  through  their  fused  compounds."  —  Newell's 
Descriptive  Chemistry, 

613.   The  Secondary  or  Storage  Cell.  —  When  a  current  is  sent 
through  an  electrolytic  cell  containing  dilute  sulphuric  acid  and 

lead  electrodes,  the 
electrolysisof  the  liquid 
liberates  oxygen  at  the 
anode  and  hydrogen  at 
the  cathode.  The  hy- 
drogen gathers  in  small 
bubbles  and  escapes. 
The  oxygen,  or  a  part 
of  it,  combines  with 
the  anode,  forming  a 
brownish  layer  of  lead 
peroxide  (PbOg)  upon 
its  surface.  When  the 
anode  is  in  this  con- 
dition, the  electrolytic 
cell  is  said  to  be 
charged^  and  is  itself 
capable  of  generating 
an  electric  current. 
This  may  be  shown 
by  disconnecting  the 
electrodes  from  the  battery  and  connecting  them  with  a  galva- 
nometer {Exp.).  It  will  be  found  that  the  direction  of  the 
current  generated  by  the  cell  is  opposite  to  the  current  by  which 


Fig.  330. 


Electrostatics  411 

it  is  charged ;  hence  the  positive  plate  of  the  cell  is  the  one 
that  receives  the  deposit  of  peroxide. 

This  experiment  illustrates  the  principle  of  the  secondary  or 
storage  cell.  If  the  electrodes  are  several  inches  square,  the  cur- 
rent will  probably  be  sufficient  to  ring  an  electric  bell  or  run  a 
small  motor ;  but  only  for  a  moment.  While  the  cell  is  generat- 
ing a  current,  hydrogen  goes  to  the  positive  plate  and,  uniting 
with  oxygen  from  the  peroxide,  reduces  it  to  monoxide  (PbO). 
At  the  same  time,  oxygen  is  liberated  at  the  negative  plate  and 
forms  a  layer  of  monoxide  upon  it.  When  the  plates  have  thus 
been  brought  to  the  same  condition,  the  cell  is  exhausted.  It  can 
be  charged  again  by  sending  a  battery  current  through  it,  as  before. 

The  plates  of  storage  cells  are  commonly  cast  in  the  form  of  a 
grating  or  grid,  the  holes  of  which  are  filled  with  a  paste  of  an 
oxide  of  lead.  This  greatly  increases  the  capacity  of  the  cell ; 
and  the  capacity  is  further  increased  by  the  use  of  a  number  of 
plates  in  each  cell,  positive  plates  alternating  with  negative  plates, 
as  shown  in  Fig.  330.  A  charging  current  from  a  dynamo  peroxi- 
dizes  the  positive  plates  and  reduces  the  oxide  of  the  negative 
plates  to  spongy  lead.  A  charged  secondary  cell,  like  an  ordinary 
or  primary  cell,  possesses  a  store  of  chemical  potential  energy. 
The  important  advantage  of  secondary  cells  is  that  the  materials 
composing  them  can  be  used  over  and  over  again  indefinitely. 
Electric  light  and  power  stations  are  operated  more  economically 
when  equipped  with  large  storage  batteries,  since  they  can  be 
charged  when  the  dynamos  are  available  for  the  purpose  and  used 
when  needed. 

X.  Electrostatics 

614.  Static  and  Current  Electricity.  —  In  the  preceding  sections 
we  have  had  to  do  only  with  electricity  in  motion,  or  ctirrent  elec- 
tricity. Under  certain  conditions  electricity  accumulates  upon  the 
surface  of  bodies  and  remains  at  rest  there  for  some  time.  In  this 
condition  it  is  0,2^^^  static  electricity  ;  and  the  study  of  the  phenom- 
ena to  which  it  gives  rise  is  called  electrostatics. 


412  Electricity 

Electrostatics  is  the  older  branch  of  electrical  science,  and  hence 
possesses  great  historical  interest.  The  experiments  are  also  inter- 
esting and  curious  to  an  exceptional  degree.  Static  electricity  is, 
however,  of  but  little  importance  compared  with  current  electricity. 

515.  Electrification  by  Friction.  —  When  a  nonconductor  or 
an  insulated  conductor  is  mbbed  with  another  substance,  it  acquires 
the  power  of  attracting  bits  of  paper  or  pith,  feathers,  and  other 
light  bodies  (Exp.).  The  strongest  effects  are  obtained  by  rub- 
bing a  vulcanite  rod  with  fur  or  flannel,  the  attractive  power  in  this 
case  being  sufficient  to  rotate  a  slender  wooden  stick  balanced  on 
a  vertical  pivot  {Exp.).  Sealing  wax  rubbed  with  furor  flannel 
and  glass  rubbed  with  silk  generally  give  good  results ;  but  in  all 
cases  the  substance  must  be  dry,  and  the  drier  the  atmosphere,  the 
better. 

It  was  known  to  the  ancient  Greeks  that  amber  possesses  this 
power  of  attraction  when  rubbed  ;  and  it  is  from  the  Greek  word 
for  amber,  eUktron^  that  the  word  eleciriciiy  was  derived  as  the 
name  of  the  agent  to  which  these  attractions  are  due.  The  process 
of  imparting  a  charge  of  electricity  to  a  body,  whether  by  friction 
or  other  means,  is  called  eUctrification^  and  the  body  is  said  to  be 
in  a  state  of  eUctrificatioriy  or  to  be  eUctrified. 

Static  electricity  differs  from  current  electricity  as  a  lake  differs 
from  a  running  stream.  When  a  charge  of  electricity  is  given  the 
opportunity  of  escaping  through  a  conductor,  it  does  so  as  an 
electric  current,  and  is  capable  of  producing  the  usual  effects  of  a 
current  in  proportion  to  its  strength  and  duration  ;  but  the  quantity 
of  electricity  in  even  a  large  static  charge  is  excessively  small  com- 
pared with  the  quantity  that  a  single  voltaic  cell  can  supply  in  a 
fraction  of  a  second. 

516.  Positive  and  Negative  Electrification ;  Attraction  and 
Repulsion.  —  An  electrified  vulcanite  rod  suspended  in  a  wire 
stirrup  by  a  thread  (Fig.  331)  turns  away  when  another  electrified 
vulcanite  rod  is  brought  near  it,  showing  that  it  is  repelled.  Two 
electrified  glass  rods  also  exhibit  repulsion  when  tested  in  the  same 
way ;  but  a  vulcanite  rod  rubbed  with  fur  and  a  glass  rod  rubbed 


Electrostatics  4 1  3 

with  silk  attract  each  other  {Exp.).  The  action  of  electrified 
bodies  is,  of  course,  mutual,  —  they  attract  each  other  equally  or 
repel  each  other  equally  (Newton's  third 
law) . 

Any  electrified  body  would  repel  either 
the  electrified  glass  or  the  electrified 
vulcanite  and  would  attract  the  other; 
hence  there  are  two  states  or  conditions 
of  electrification.  Electrification  like 
that  of  glass  when  rubbed  with  silk  is 

called  positive  electrification  ;  electrification  like  that  of  vulcanite 
when  rubbed  with  fur  or  flannel  is  called  negative.  Two  bodies 
repel  each  other  if  both  are  positively  or  both  negatively  electrified; 
they  attract  each  other  if  oppositely  electrified. 

Different  theories  have  been  advanced  concerning  the  nature  of 
electricity  and  the  difference  between  positive  and  negative  elec- 
trification ;  but  such  questions  are  still  undecided  and  are  not  con- 
si  lered  in  this  book.  It  should  be  noted  that  the  terms  positive 
and  negative  electrification  refer  to  unlike  electrical  states  or  con- 
ditions^ and  do  not  imply  that  there  are  two  kinds  of  electricity. 
Experiment  shows  that,  when  oppositely  electrified  bodies  are 
connected  by  a  conductor,  the  direction  of  the  current,  as  the  term 
is  used  in  current  electricity  (Art.  442,  third  paragraph),  is  from 
the  positively  electrified  to  the  negatively  electrified  body. 

517.  Electrification  by  Contact  with  an  Electrified  Body.  —  The 
bits  of  paper  or  other  light  bodies  that  cling  to  an  electrified  rod 
often  dart  away  after  contact  (Exp.).  The  explanation  of  this 
behavior  is  that,  by  contact  with  the  rod,  the  pieces  of  paper 
receive  a  portion  of  its  charge  and  are  then  repelled,  since  they 
are  in  the  same  state  of  electrification  as  the  rod.  Since  the  rod 
is  a  nonconductor,  it  parts  with  its  charge  very  slowly,  even  at 
points  of  actual  contact  with  the  pieces  of  paper.  This  explains 
why  some  of  the  pieces  are  not  repelled. 

518.  Electrostaticlnduction. — When  an  electrified  rod  is  brought 
near  an  unelectrified  pith  ball  suspended  by  a  silk  thread,  the  ball 


414  Electricity 

is  attracted  and  drawn  far  out  from  the  vertical,  although  it  is  not 
permitted  to  touch  the  rod  {Exp.).     The  cause  of  the  attraction  is 
illustrated  in   Fig.  332,  assuming  the  charge  of  the  rod   to  be 
negative.    The  pith  ball  is  perfectly  insulated  by 
the  silk  thread,  but  is  itself  a  conductor.     Now 
it  can  be  shown  by  experiment  that  the  presence 
of  an  electrified  body  near  an  unelectrified,  insu- 
lated conductor  causes  a  charge  of  the  opposite 
sign  (unlike  charge)  on  the  nearer  side  of  the 
Fig.  33a.         conductor,  and  an  equal  charge  of  the  same  sign 
on  its  farther  side.    This  action  is  called  electro- 
static induction^  and   the   resulting   charges   are  called  induced 
charges.    The  nearer  side  of  the  pith  ball  is  attracted  by  the  rod, 
their  charges  being  unlike ;   the  farther  side  of  it  is  repelled,  but 
with  a  less  force,  since  the  distance  is  greater.     Hence  the  resultant 
force  is  an  attraction. 

It  is  instructive  to  compare  electrostatic  induction  with  mag- 
netic induction.  Recall  the  fact  that  a  piece  of  soft  iron  is 
attracted  by  a  magnet  because  it  is  first  magnetized  by  the  pres- 
ence of  the  magnet.  The  electric  attraction  of  an  unelectrified 
body  is  always  due  to  induction.  If  the  attracted  body  is  not 
insulated,  the  repelled  charge  escapes,  as  shown  in  later  experi- 
ments. 

519.  Permanent  Electrification  by  Induction.  —  The  equal  and 
opposite  charges  induced  on  the  pith  ball  in  the  preceding  experi- 
ment are  temporary;  />.  they  continue  only  so  long  as  the  pres- 
ence of  the  inducing  charge  keeps  them  separated.  But  if  the 
ball  be  touched  by  the  finger  while  a  negatively  electrified  rod  is 
held  near  it,  the  negative  induced  charge  escapes,  leaving  only  a 
positive  charge  on  the  ball.  This  remains  as  a  permanent  induced 
charge  when  the  finger  is  removed  before  the  rod  is.  The  pres- 
ence of  a  positive  charge  on  the  ball  is  proved  if  the  ball  is 
repelled  on  bringing  up  an  electrified  glass  rod. 

It  will  be  seen  that  a  permanent  charge  obtained  by  induction 
is  unlike  the  inducing  charge. 


Electrostatics  415 

520.  The  Electroscope.  —  The  suspenled  pith  ball  can  be  used 
as  an  electroscope  to  detect  the  presence  of  a  charge  upon  any 
body  and  to  determine  its  sign.  The  pith  ball  is  given  a  charge 
of  known  sign,  and  the  body  to  be  tested  is  brought  near  it.  If  the 
ball  is  repelled,  the  body  is  electrified,  and  its  charge  is  like  that 
of  the  ball.  The  attraction  of  the  ball  is  not  a  certain  test  of 
electrification.     (Why  not?) 

521.  The  Electric  Field.  —  Electrostatic  forces,  like  magnetic 
forces,  are  supposed  to  act  through  the  medium  of  the  ether.  The 
two  forces  are  of  a  different  character,  however,  and  are  wholly 
unrelated;  for  static  electricity  and  magnetism  have  no  effect 
upon  each  other.  The  terms  electric  field  and  lines  of  electric  force 
correspond  respectively  to  magnetic  field  and  lines  of  magnetic 
force. 

522.  Electrical  Machines.  —  Machines  for  developing  and  col- 
lecting charges  of  electricity  are  of  two  types ;  namely,  friction 
machines  and  induction  machines. 

Friction  Machines.  —  Figure  333  represents  one  form  of  friction 
machine.     A  positive  charge  is  developed  on   a   large  revolving 
glass  disk,  A^  by  the 
friction  of  leather  pads, 
B.    The  charge  is  col- 
lected on  each  side  by 

a    number    of    points  ^        ^^i^^n^jj^^  /\  Iw 

which  project  from  a 
brass  rod  and  nearly 
touch  the  disk.  The 
rods  carry  the  charge 
to   an  insulated   brass  yig.  333. 

cylinder,  C,  from  which 

it  can  be  drawn  off  as  a  spark  discharge  by  bringing  the  finger  or 
any  other  conductor  near  it.  A  spark  a  centimeter  or  more  in 
length  can  be  obtained  in  this  manner  from  a  machine  in  good 
condition. 

Friction  machines  of  various  forms  were  invented  during  the 


4i6 


Electricity 


eighteenth  century;    but   they  are  greatly  inferior  to  the  more 

modern  induction  machines,  by  which  they  have  been  superseded. 

Induction   Machines, — The   elecirophorus   (Fig.   334)    is    the 

simplest  induction  machine.     It  consists  of  a  disk  of  vulcanite 

or  other  resinous  material,  and 
a  metal  disk  of  slightly  smaller 
diameter,  provided  with  an  insu- 
lating handle.  The  vulcanite  is 
negatively  electrified  by  striking 
or  rubbing  it  with  catskin  or 
flannel,  and  the  metal  disk  or 
cover  is  then  placed  upon  it.  The 
disks  really  touch  at  only  a  few 
points ;  and,  as  the  vulcanite  is  a 
nonconductor,  it  does  not  lose  any  appreciable  part  of  its  charge 
to  the  cover.  The  entire  charge,  however,  acts  inductively  on  the 
cover,  producing  a  positive  charge  on  its  lower  side  and  a  nega- 
tive charge  on  its  upper  side  (Fig.  335,  A).  The  negative  charge 
is  repelled  by  the  charge  on  the  vulcanite,  and  is  permitted  to 
escape  by  touching  the  cover  with  the  finger.  The  positive 
charge  remains,  being  "  bound  "  by  the  attraction  of  the  charge 
on  the  vulcanite  (Fig.  335,  B).     This  leaves  the  disk  positively 


Fig.  334, 


cs 


Fig.  335. 


electrified  when  it  is  lifted  from  the  vulcanite  by  means  of  llie 
handle.  It  can  then  be  discharged  by  bringing  the  finger  or  other 
conductor  near  it,  as  its  charge  is  no  longer  bound.  The  cover 
can  be  repeatedly  charged  and  discharged  in  this  manner  without 
again  rubbing  the  vulcanite. 


Electrostatics 


417 


Different  forms  of  induction  machines  have  been  invented 
which  are  wholly  automatic  in  their  action,  and  are  much  more 
powerful  than  the  electrophorus.  One  of  these  —  the  Toepkr-Holtz 
machine  —  is  shown  in  Fig.  336.  The  movable  parts  are  carried 
on  a  large  glass  disk.  Positive  and  negative  charges  are  induced 
on  different  parts  of  the  revolving  disk  as  they  pass  certain  points. 


These  charges  are  collected  by  projecting  metallic  points,  as  in 
the  friction  machine,  and  accumulate  on  insulated  conductors 
until  the  difference  between  their  potentials  is  sufficient  to  cause 
a  spark  discharge  across  the  gap  between  the  knobs.  Sparks  from 
5  cm.  to  10  cm.  long  can  be  obtained  from  machines  of  moder- 
ate size.  Full  descriptions  of  these  machines  are  to  be  found  in 
larger  works. 


41 8  Electricity 

523.  Potential  of  SUtic  Electricity.  —  The  same  diffcience  of 
potential  is  required  to  produce  a  spark  of  given  length  whether  it 
is  obtained  from  an  electrified  body  or  an  induction  coil.  Hence 
we  know  from  the  lengths  of  the  sparks  that  the  potentirls  due  to 
electrification  are  often  from  tens  of  thousands  to  hundreds  of 
thousands  of  volts.  A  potential  difference  of  100,000  to  300,000 
volts  can  be  obtained  with  a  Toepler-Holtz  machine  of  medium 
size.  In  fact,  the  potential  to  which  it  is  possible  to  charge 
the  machine  or  any  body,  depends  only  upon  the  insulation  and 
the  dryness  of  the  atmosphere.  Beyond  this  limit  the  charge 
escapes  as  rapidly  as  it  is  developed  or  imparted.  Charges  at 
even  the  highest  potentials  mentioned  are  not  dangerous  unless 
the  quantity  of  the  charge  is  much  larger  than  is  generally  the  case. 

An  electrical  machine  is  capable  of  furnishing  small  quantities 
of  electricity  in  the  form  of  an  intermittent  current  of  very  high 
E.  M.  F.,  like  the  current  produced  by  an  induction  coil  (Art.  499). 
Most  experiments  requiring  a  high-potentiaU  current  can  there- 
fore be  performed  either  with  af  machine  or  a  coil. 

524.  The  Leyden  Jar.  —  The  Leyden  jar  (Fig.  337)  is  a  device 
for  accumulating  and  storing  a  large  charge,  either  positive  or 

negative.  It  received  its  name  from  the  city  of  Leyden 
in  the  Netherlands,  where  it  was  invented  in  1746. 
It  consists  of  a  glass  jar  coated  inside  and  out  for 
about  two  thirds  its  height  with  tin  foil.  A  brass  rod, 
terminating  in  a  knob  at  the  top,  extends  through  the 
stopper,  and  is  connected  with  the  inner  coat  of  the  jar 
by  means  of  a  chain  attached  to  its  lower  end.  The 
jar  is  charged  by  connecting  the  rod  or  the  knob  with 
one  of  the  terminals  of  an  electrical  machine.  To  dis- 
'  charge  it,  one  end  of  a  short  conductor  is  touched  to 

its  outer  coat  and  the  other  end  brought  near  the  knob.  When 
it  is  sufficiently  near,  a  spark  occurs,  discharging  the  jar.  The 
discharging  conductor  must  be  provided  with  an  insulating  handle 
to  avoid  a  shock  which,  with  a  large  jar,  would  be  very  painful  and 
possibly  dangerous. 


Electrostatics  419 

The  action  of  the  jar  is  illustrated  by  the  following  experiments : 
(i)  The  jar  is  placed  on  the  table  and  charged,  by  means  of  an 
induction  machine,  for  a  certain  length  of  time  or  until  the  handle 
of  the  mafchine  has  made  a  certain  number  of  revolutions.  The 
jar  is  then  discharged,  and  the  length  and  intensity  of  the  spark 
noted.  (2)  The  jar  is  charged  for  the  same  length  of  time  as 
before,  while  standing  on  an  insulator  (a  large  sheet  of  vulcanite 
or  glass),  and  again  discharged.  The  spark  obtained  with  this 
discharge  fs  much  thinner  and  less  brilliant  than  before,  indicating 
that  the  quantity  of  electricity  in  the  charge  is  much  less  (^Exp.).. 
The  explanation  of  this  difference  is  as  follows :  While  the  charge 
is  accumulating  on  the  inner  coat  of  the  jar  in  the  first  experi- 
ment, it  attracts  an  opposite  charge  to  the  outer  coat  by  induction 
through  the  glass.  In  the  accumulation  of  this  charge  the  table 
serves  as  a  conductor.  In  the  second  experiment  the  insulation 
of  the  jar  prevents  an  induced  charge  on  the  outer  coat.  When 
the  outer  coat  of  the  jar  is  not  insulated,  the  charge  that  is  induced 
on  it  attracts  the  charge  on  the  inner  coat.  This  attraction  de- 
creases the  potential  of  the  latter  charge,  and  consequently  increases 
the  rate  at  which  the  charging  takes  place.  It  follows  that  the 
attraction  of  the  induced  charge  increases  the  amount  of  the 
charge  on  the  inner  coat  for  a  given  potential ;  or,  in  other 
words,  the  induced  charge  increases  the  capacity  of  the  condenser. 

The  extent  to  which  the  capacity  of  a  jar  is  thus  increased  is 
perhaps  best  shown  by  the  action  of  the  jars  of  an  induction 
machine  (Fig.  ^2>^)'  When  the  machine  is  operated,  the  charges 
accumulate  principally  in  the  jars,  the  positive  charge  in  one,  the 
negative  in  the  other.  The  outer  coats  of  the  jars  become  oppo- 
sitely charged  by  induction,  each  receiving  its  charge  from  the 
other  through  a  metal  conductor  by  which  they  are  connected 
under  the  base  of  the  instrument.  Under  these  conditions,  the 
machine  gives  a  thick,  brilliant  spark  at  intervals  of  several 
seconds.  When  the  outer  coats  of  the  jars  are  disconnected 
by  opening  a  switch  (not  shown  in  the  figure),  the  sparks  are 
thin  and  much  less  brilliant  than  before,  and,  at  the  same  time, 


420  Electricity 

are  much  more  frequent  {Exp.),  The  quantity  of  electricity  that 
is  discharged  with  each  spark  is  evidently  very  much  less  than 
before,  indicating  a  corresponding  decrease  in  the  capacity  of  the 
jars.  This  is  due  to  the  fact  that  the  outer  coats  of  the  jars  do 
not  become  charged,  the  wooden  base  of  the  machine  being 
practically  an  insulator  for  such  rapid  action. 

The  Leyden  jar  is  one  form  of  condenseKf  the  essential  parts 
of  a  condenser  being  two  conductors  very  near  each  other,  but 
separated  by  an  insulator.  A  sheet  of  glass,  mica,  or  paraffined 
paper,  with  a  smaller  sheet  of  tinfoil  attached  to  each  side,  leav- 
ing a  wide  margin  of  the  insulator  round  the  foil,  is  a  simple  form 
of  condenser. 

525.  Distribution  of  an  Electrical  Charge ;  Effect  of  Points.  — 
It  is  shown  by  experiments  not  to  be  described  here  that  an  elec- 
trical charge  resides  wholly  upon  the  surface  of  a  solid  body,  and 
only  upon  the  outer  surface  of  a  hollow  body  unless  there  is  an 
opposite  charge  upon  another  body  inside  it.  This  is  due  to  the 
repulsion  of  all  parts  of  a  charge  for  every  other  part. 

If  the  charged  body  is  a  conductor,  this  self-repulsion  of  the 
charge  causes  a  definite  distribution  of  it,  which  depends  only  upon 
the  shape  of  the  conductor  (assuming  that  there  are  no  other 
charges  in  the  vicinity  to  cause  induction).  A  charge  is  distrib- 
uted uniformly  over  the  surface  of  a  sphere ;  upon  other  bodies 
the  quantity  per  unit  of  surface,  or  the  electric  density,  is  greater 
where  the  curvature  is  greater,  and  is  greatest  at  edges,  corners, 
and  especially  at  points. 

"  At  a  point,  indeed,  the  density  of  the  collected  electricity  may 
be  so  great  as  to  electrify  the  neighboring  particles  of  air,  which 
then  are  repelled,  thus  producing  a  continual  loss  of  charge.  For 
this  reason  points  and  sharp  edges  are  always  avoided  on  electrical 
apparatus,  except  where  it  is  specially  desired  to  set  up  a  discharge. 
The  effect  of  points  in  discharging  electricity  from  the  surface  of  a 
conductor  may  be  readily  proved  by  numerous  experiments.  If  an 
electrical  machine  be  in  good  working  order,  and  capable  of  giving, 
say,  sparks  four  inches  long  when  the  knuckle  is  presented  to  the 


Electrostatics 


421 


Fig.  338. 


knob,  it  will  be  found  that,  on  fastening  a  fine-pointed  needle  to  the 

conductor,  it  discharges  the  electricity  so  effectually  at  its  point  that 

only  the  shortest  sparks  can 

be  drawn  at  the  knob,  while  a 

fine  jet  or  brush  of  pale  blue 

light  will  appear  at  the  point. 

If  a  lighted  taper  be  held  in 

front  of  the  point,  the  flame 

will   be   visibly  blown   aside 

(Fig.  338)  by  the  streams  of 

electrified  air  repelled  from 

the  point.    These  air  currents 

can  be  felt  with  the   hand. 

They  are  due  to  a  mutual  repulsion  between  the  electrified  air 

particles  near  the  point  and  the  electricity  collected  on  the  point 

itself."  —  S.  P.  Thompson's  Elementary  Lessons  in  Electricity  and 

Magnetisfn. 

526.    The  Electric  Discharge  in  Rarefied  Gases. — The  electric 

discharge  is  produced  in  rarefied  gases  by  means  of  glass  tubes  or 

bulbs,  provided  with  electrodes  of  platinum  wire  fused  into  the 

glass.     Such  tubes,  when  exhausted  to  a  pressure  of  one  or  two 

millimeters  of  mer- 
cury, are  known  as 
Geissler  tubes  (  Fig. 
339).  The  differ- 
ence of  potential 
required  to  pro- 
duce   a    discharge 


s^ 


\^-^. 


Fig.  339. 


in  any  gas,  between 


electrodes  at  a  given  distance  apart,  decreases  as  the  pressure  of 
the  gas  is  diminished ;  and  an  induction  coil  giving  a  spark  i  cm. 
long  in  air  will  illuminate  a  Geissler  tube  12  or  15  cm.  long.  An 
induction  machine  can  also  be  used  for  the  purpose. 

"Through   such    tubes,    before   exhaustion,    the   spark   passes 
without  any  unusual  phenomena  being  produced.     As  the  air  is 


422  Electricity 

exhausted,  the  sparks  become  less  sharply  defined,  and  widen  out 
to  occupy  the  whole  tube,  becoming  pale  in  tint  and  nebulous 
in  form.  The  cathode  exhibits  a  beautiful  bluish  or  violet  glow, 
separated  from  the  conductor  by  a  narrow  dark  space^  while  at  the 
anode  a  single  small  bright  star  of  light  is  all  that  remains.  At  a 
certain  degree  of  exhaustion  the  light  in  the  tube  breaks  up  into  a 
set  of  striip,  or  patches  of  light  of  a  cup-like  form,  which  vibrate  to 
and  fro  between  darker  spaces."  (Thompson.)  The  color  of  the 
discharge  in  Geissler  tubes  is  different  with  different  gases.^ 

527.  Atmospheric  Electricity.  —  The  sparks  obtained  from 
electrical  machines  and  Leyden  jars  suggested  to  a  number  of  the 
early  experimenters  in  electricity  that  lightning  was  due  to  elec- 
trical discharges  in  the  atmosphere.  Benjamin  Franklin  put  this 
theory  to  an  experimental  test  in  1752.  "  He  sent  up  a  kite  dur- 
ing the  passing  of  a  storm,  and  found  the  wetted  string  to  conduct 
electricity  to  the  earth,  and  to  yield  abundance  of  sparks.  These 
he  drew  from  a  key  tied  to  the  string,  a  silk  ribbon  being  interposed 
between  his  hand  and  the  key  for  safety.  Leyden  jars  could  be 
charged,  and  all  other  electrical  effects  produced,  by  the  sparks 
furnished  from  the  clouds.  The  proof  of  the  identity  was  com- 
plete."    (Thompson.) 

It  has  been  repeatedly  shown  by  later  experiments  that  the 
atmosphere  is  generally  electrified  even  in  fair  weather.  In  fair 
weather  the  electrification  is  almost  always  positive ;  in  stormy 
weather  it  is  sometimes  positive  and  sometimes  negative.  The 
potential  increases  with  the  altitude ;  but  differs  widely  in  different 
localities  and  with  different  states  of  the  weather.  The  rise  of 
potential  has  been  found  as  great  as  600  volts  per  meter  of  eleva- 
tion above  the  ground. 

1  As  the  exhaustion  in  a  vacuum  tube  is  continued  beyond  a  pressure  of  one 
millimeter,  the  dark  space  surrounding  the  cathode  increases  in  width  u:;;!).  >Jicn 
the  pressure  is  reduced  to  about  one  millionth  of  an  atmosphere,  it  completely  fills 
the  tube.  Tubes  exhausted  to  this  degree  are  called  Crookes  tubes.  The  eloctric 
discharge  in  a  Crookes  tube  produces  new  forms  of  radiation,  called  cathode  rays 
and  X-rays.  The  latter  are  also  known  as  Roentgen  rays,  from  their  discoverer. 
The  pupil  is  referred  to  other  works  for  an  account  of  these  rays  and  their  applications. 


Electrostatics  423 

Various  theories  have  been  advanced  to  account  for  the  electri- 
fication of  the  atmosphere ;  but  very  little  is  definitely  known 
about  it.  Evaporation  is  very  probably  one  of  the  principal  causes. 
But,  aside  from  this  question,  if  we  suppose  the  particles  of  water 
vapor  in  the  atmosphere  to  be  even  slightly  electrified,  the  high 
potential  of  clouds  is  easily  explained ;  for,  as  the  particles  unite 
in  the  process  of  condensation,  the  charge  increases  much  more 
rapidly  than  the  capacity  of  the  drop.  For  example,  if  one  million 
equally  charged  particles  unite,  the  potential  becomes  ten  thousand 
times  as  great ;  and  there  are  thousands  of  milHons  of  particles  in 
a  drop.  The  great  length  of  lightning  sparks  or  flashes  shows 
that  the  potential  of  a  thunder-cloud  is  enormously  high. 

528.  Thunder. — Thunder  corresponds  to  the  snapping  sound 
produced  by  an  electric  spark.  The  sudden  heating  of  the  air 
along  the  path  of  a  lightning  flash  causes  it  to  expand  with  explo- 
sive violence,  producing  sound  waves  of  great  intensity.  If  the 
flash  is  short  and  straight,  the  sound  is  a  short  clap ;  if  it  is  long 
and  zigzag,  the  sounds  produced  by  its  different  parts  have  unequal 
distances  to  travel  to  the  observer  and  are  heard  in  quick  succes- 
sion as  a  continuous  rattle.  The  rolling  sound  of  distant  thunder 
is  due  to  various  reflections  of  the  sound  from  clouds,  from  the 
ground,  and  often  from  neighboring  hills. 

529.  Lightning  Conductors. — The  use  of  lightning  conductors 
to  protect  buildings  was  first  suggested  by  Benjamin  Franklin. 
The  usual  device  consists  of  one  or  more  iron  rods,  extending 
some  distance  above  the  highest  points  of  a  building  and  connected 
by  means  of  large  iron  or  copper  conductors  with  datnp  earth  or, 
better,  with  water.  If  the  conductor  ends  in  dry  earth,  it  is  not 
only  useless  but  even  dangerous.  (Why?)  Each  rod  is  terminated 
by  a  gilded  copper  point. 

The  action  of  a  lightning  conductor  depends  largely  upon  in- 
duction. A  charged  cloud  induces  an  opposite  charge  on  the 
ground  under  it  and  on  houses,  trees,  and  other  objects  within 
this  area.  This  inductive  action  is  strongest  upon  the  highest 
objects,  and  causes  lightning  rods  to  become  highly  electrified. 


424 


Electricity 


Under  these  conditions  a  rapid  and  continuous  discharge  takes 
place  from  the  sharply  pointed  tips  of  the  rods  (Art.  525).  This 
quiet  discharge  of  opposite  electrification  toward  the  cloud  is 
often  sufficient  to  prevent  lightning ;  but,  if  a  stroke  does  occur, 
the  rod  receives  the  discharge  and  the  building  is  preserved. 

530.  The  Aurora  Borealis.  —  "In  the  northern  regions  of  the 
earth  the  aurora  horfalis,  or  northern  lights^  is  an  occasional  phe- 
nomenon ;  and  within  and  near  the  Arctic  Circle  is  of  almost 
nighdy  occurrence.  Similar  lights  are  seen  in  the  south  polar 
regions  of  the  earth,  and  are  denominated  aurora  ausiralis.    As 


Fig.  34a 

seen  in  European  latitudes,  the  usual  form  assumed  by  the  aurora 
is  that  of  a  number  of  ill-defined  streaks  or  streamers  of  a  pale 
tint  (sometimes  tinged  with  red  and  other  colors),  either  radiat- 
ing in  a  fanlike  form  from  the  horizon  in  the  direction  of  the 
magnetic  north,  or  forming  a  sort  of  arch  across  that  region  of 
the  sky,  of  the  general  form  shown  in  Fig.  340.  A  certain  flicker- 
ing or  streaming  motion  is  often  discernible  in  the  streaks.  Under 
very  favorable  circumstances  the  aurora  extends  over  the  entire 


Electrostatics  425 

sky.  The  appearance  of  an  aurora  is  usually  accompanied  by  a 
magnetic  storm,  affecting  the  compass-needles  over  whole  regions 
of  the  globe.  This  fact,  and  the  position  of  the  auroral  arches 
and  streamers  with  respect  to  the  magnetic  meridian,  directly 
suggest  an  electric  origin  for  the  light,  —  a  conjecture  which  is 
confirmed  by  the  many  analogies  between  auroral  phenomena 
and  those  of  discharge  in  rarefied  air.  Yet  the  presence  of  an 
aurora  does  not,  at  least  in  our  latitudes,  affect  the  electric  condi- 
tions of  the  lower  regions  of  the  atmosphere. 

"  The  most  probable  theory  of  the  aurora  is  that  originally  due 
to  Franklin;  namely,  that  it  is  due  to  electric  discharges  in  the 
upper  air,  in  consequence  of  the  differing  electrical  conditions 
between  the  cold  air  of  the  polar  regions  and  the  warmer  streams 
of  air  and  vapor  raised  from  the  level  of  the  ocean  in  tropical 
regions  by  the  heat  of  the  sun."     (Thompson.) 


APPENDIX 


Table  I 


Metric  Units  of  Lengthy  Surface^  and  Volume 


lo  millimeters  (mm.) 
lo  centimeters  (cm.) 
lo  decimeters  (dm.) 
loo  sq.  millimeters  (smm.) 
lOo  sq.  centimeters  (scm.) 
loo  sq.  decimeters  (sdm.) 
looo  cu.  millimeters  (cmm.) 
looo  cu.  centimeters  (ccm.) 
xooo  cu.  decimeters  (cdm.) 


:  I  centimeter 

:  I  decimeter 

:  I  meter  (m.) 

:  I  sq.  centimeter 

:  I  sq.  decimeter 

:  I  sq.  meter 

1  cu.  centimeter 

:  I  cu.  decimeter 

:  I  cu.  meter  (cu.  m.) 


Table  n 


Equivalents 


Metric  to  English 

cm.     =   .3937  in. 
m-        =39-37  in. 
km.     =.6214  mile 


scm.    =.1550  sq.  in. 

sq.  m.  =  1.196  sq.  yd. 

=  10.764  sq.  ft. 

I  ccm.    =.06103  cu.  in. 
I  cdm.    =1.0567  qt.  (liquid) 
I  cu.  m.=  1.308  cu.  yd. 
=  35-317  cu.  ft 


English  to  Metric   * 

I  in.        =  2.540  cm. 

I  ft.         =  30.48  cm. 

I  yd.       =.9144  m. 

I  mile     =  1.6093  km. 

I  sq.  in.  =6.452  scm. 

I  sq.ft.   =929.0  scm. 

I  sq.  yd.  =  .8361  sq.  m. 

I  cu.  in.  =  16.387  ccm. 

I  cu.  ft.  =28,315.  ccm. 

I  cu.  yd.  =  .7645  cu.  m. 

I  qt.        =  .9463  cdm.  (liters) 

I  gal.       =  3.785  liters 


I  gram   =  .0353  oz. 
I  kg.       =  2.2046  lb. 


426 


oz. 
lb. 


=  28.35  g. 
=  453.6  g. 


Appendix 


427 


Table  III 

Mensuration  Rules 

ratio  of  the  circumference  of  a  circle  to  its  diameter  =  3.1416 

Circumference  of  a  circle  (radius,  r)  =  2  Trr 

Area  of  a  circle  =  -nr^ 

Surface  of  a  sphere  =  4  irr^ 

Volume  of  a  sphere  =  f  Trr^ 
Lateral  surface  of  a  right  cylinder 

(altitude  h  and  radius  of  base  r)  =  2  irrh 

Volume  of  a  right  cylinder  =  irr^h 


Table  IV 
Densities  (in  grams  per  ccm.) 


Aluminum  .  . 
Antimony,  cast  . 
Beeswax  .  .  . 
Bismuth,  cast  . 
Brass  .... 
Copper  .  .  . 
Cork  .  .  .  . 
Galena  .  .  . 
German  silver  . 
Glass,  crown  .  . 
Glass,  flint  .  . 
Gold    .     .     .     . 

Ice 

Iron,  bar  .  .  . 
Iron,  cast .  .  . 
Ivory  .  .  .  . 
Lead  .  .  .  . 
Marble  .  .  . 
Mercury,  at  o°C. 
Platinum  .  .  . 
Quartz  .  .  . 
Silver  .  .  .  . 
Steel  .  .  .  . 
Sulphur,  native  . 
Tin.  ...  . 
Zinc,'  cast .     .     . 


2.67 
6.72 

.96 
9.8 
8.4 
8.8    to 

.14  to 
7.58 
8.5 

2-5 

3       to 

19-3 
.917 
7.8 
7.2    to 

1.9 

11.3    to 

2.72 

13-596 
21.5 
2.65 

10.4 

7.8 

2.03 

7-3 
6.86 


8.9 
.24 


3-5 


7.3 


11.4 


to  10.5 
to    7.9 


Alcohol  (95%)  .     .       .82 

Blood 1.06 

Carbon  disulphide  .     1.29 
Chloroform    .     .     .     1.5 
Copper  sulphate  solution  i .  1 6 

Ether 72 

Glycerine  .  .  .  .  1.27 
Hydrochloric  acid  .  1.22 
Mercury,  at  0°  C.    .   13.596 

Milk 1.03 

Nitric  acid     ...     1.5 
Oil  of  turpentine     .       .87 

Olive  oil 915 

Sulphuricacid(i5%)     i.io 
Sulphuric  acid     .     .     1.8 
Water  (4°  C.)     .     .     i.ooo 
Water,  sea     .     .     .     1.026 

Gases  at  0°  C.  and  76  cm. 
Pressure 

Air 001293 

Carbon  dioxide  .     .       .001977 
Hydrogen      .     .     .       .0000896 

Nitrogen 001256 

Oxygen 001430 


428 


Appendix 


Table  V 

Tangents  of  Angles 

To  find  the  tangent  of  an  angle  not  measured  by  a  whole  num- 
ber of  degrees,  find  first  the  tangent  of  the  integral  part  of  the 
number,  and  add  to  this  the  product  obtained  by  multiplying  the 
difference  between  this  tangent  and  the  tangent  of  the  next  whole 
number  of  degrees  by  the  decimal  part  of  the  angle.  For  example, 
to  find  the  tangent  of  38°.7,  proceed  thus :  — 

tan  38*  =.781,  tan  39*  =  .810. 

.810— .781  =  .029. 

.7  X  .029  =  .020. 

tan  38^.7  =  .781  -f  .020  =  .801. 


Anglk 

Tangent 

Angle 

Tangent 

Angle 

Tangent 

Angle 

Tangent 

0° 

.0000 

23° 

.424 

46^ 

1.036 

69° 

2.61 

I 

.0175 

24 

.445 

47 

1.072 

70 

2.75 

2 

.0349 

25 

.466 

48 

I. Ill 

71 

2.90 

3 

.05  24 

26 

.488 

49 

1. 150 

72 

3.08 

4 

.0699 

27 

.510 

50 

1. 192 

73 

3-27 

5 

.0875 

28 

•532 

51 

1-235 

74 

3-49 

6 

.1051 

29 

•554 

52 

1.280 

75 

3-73 

7 

.1228 

30 

•577 

53 

1-327 

76 

4.01 

8 

.1405 

31 

.601 

54 

1.376 

77 

4-33 

9 

.1584 

32 

.625 

55 

1.428 

78 

4.70 

ID 

.1763 

33 

.649 

56 

1.483 

79 

5.14 

II 

.194 

34 

.675 

57 

1.540 

80 

5.67 

12 

•213 

35 

.700 

58 

1.600 

81 

6.31 

13 

.231 

36 

.727 

59 

1.664 

•82 

7.12 

14 

.249 

37 

.754 

60 

1-732 

S3 

8.14 

'5 

.268 

38 

.781 

61 

1.804 

84 

9-51 

16 

.287 

39 

.810 

62 

1.88 

85 

11-43 

17 

.306 

40 

•839 

63 

1.96 

86 

14.30 

18 

•325 

41 

.869 

64 

2.05 

87 

19.08 

19 

.344 

42 

.900 

65 

2.14 

88 

28.64 

20 

.364 

43 

.933 

66 

2.25 

89 

57-29 

21 

.384 

44 

.966 

67 

2.36 

90 

00 

22 

.404 

45 

1,000 

68 

2.48 

Appendix 


429 


VI  —  References  to  Chute's  Physical  Laboratory  Manual 

The  following  list  of  references  to  the  revised  edition  of  Chute's 
Physical  Laboratory  Manual,  published  by  D.  C.  Heath  &  Co., 
is  provided  for  the  convenience  of  teachers  who  may  wish  to  use 
this  manual  in  connection  with  the  text.  The  numbers  in  the 
first  column  refer  to  articles  in  the  text ;  those  in  the  last  column, 
to  articles  in  the  manual. 


Text 

14,15 

15 

15 

19 
22-25 

30-32 

35 

36 

47 

53 

62,  63 

67,68 

96,97 

119-123 

130-138 

155 

159 

162-164 

166 

190 

199 

231 

233 
235 
237 
241 
241 

245 
249 
257 
263 


Topic  illustrated  Manual 

Extension 19-21 

Capacity 25 

Volume  by  displacement ...  28 

Weighing.     .         31 

Liquid  pressure 52 

Buoyancy  of  liquids    .     .     .     .  55 

Specific  gravity  of  solids      .     .  56,  57 

Specific  gravity  of  liquids      .     .  58 

Boyle's  Law 53 

The  siphon 54 

Concurrent  forces 39 

Parallel  forces 40 

Uniformly  accelerated  motion    .  42 

Curvilinear  motion 41 

The  pendulum 43 

The  lever 44 

The  wheel  and  axle     ....  46 

Pulleys 45 

The  inclined  plane      ....  47 

Tenacity 35 

Capillarity 51 

The  fixed  points  of  a  thermometer  85 

Coefficient  of  linear  expansion   .  86 

Coefficient  of  cubical  expansion  87 

Law  of  Charles      .....  88 

Specific  heat  of  a  solid    ...  97 

Specific  heat  of  a  liquid  ...  98 

Melting  points 89 

Heat  of  fusion  of  ice       ...  loi 

Dew-point 91 

Boiling  points 90 


430 


Appendix 


Text 
267 

3»3. 

313 

337 

339 

344 

347 

351-355 

357 

362 

375-380 
382 

394 

396,  397 

402 

404 

421 

421 

426 

432 

454-456 

446 

466 

468 

471 

471 

475 

477 

478 

4781  485 
492, 495 


Topic  illustrated  Manual 

Heat  of  vaporization  of  water       .  102 

Laws  of  vibrating  strings     ...  63 

Resonance  (velocity  ol  sound)      .  59,  60 

Vibration  rate  of  a  tuning  fork      .  62 

Pinhole  images 64, 65 

Photometry 66 

Reflection  of  light 67 

Images  in  plane  mirrors  ....  69 

Concave  mirrors  .     .     .     .     .     .  70,  72,  73 

Convex  mirrors 7i>  73 

Index  of  refraction 74.  75 

Convex  lens 76,  78,  79 

Concave  lens        77>  78,  79 

The  simple  microscope  ....  80 

The  telescope 80 

Spectra 81 

Wave  length 82  • 

The  poles  of  a  magnet  ....  103 

Distribution  of  magnetic  action    .  106 

Magnetic  transparency  .     .     .     .  104 

Magnetic  fields 107 

Magnetic  eflect  of  a  current     .     .  109 

Electro-motive  series      .     .     .     .  110 

The  tangent  galvanometer  .     .     .  123,  124 
Changeof  resistance  with  temperature  1 20 

Electrical  resistance 118 

The  resistance  of  a  cell  ....122 

Fall  of  potential  along  a  conductor  125 

Resistance  of  conductors  in  parallel  119 

Electro-motive  force  of  cells    .     .  126 

Use  of  voltmeter  and  ammeter     .  121 

Induced  currents 128,129 


INDEX 


The  Numbers  refer  to  Pages 


Aberration,  chromatic,  333. 

spherical,  282,  304. 
Absolute  temperature,  180-181. 
Absorption,  of  radiation,  168,  171. 

selective,  171,  327-329. 
Acceleration,  71-74- 

due  to  gravity,  76-78. 
Accommodation,  power  of,  309. 
Achromatic  lens,  334. 
Action  and  reaction,  8,  9,  89,  340, 413. 
Adhesion,  149-150. 
After-images,  332. 
Air,  buoyancy  of,  46-48. 

composition  of,  3. 

density  of,  31. 

water  vapor  in,  1 96-197. 
Air  pump,  41-42. 
Amalgamating  zinc,  358. 
Ammeter,  384. 
Ampere,  377. 

Analysis,  of  light,  322,  327. 
Angle,  critical,  293. 

of  deviation,  284,  291. 

of  incidence  and  reflection,  269. 

of  refraction,  284. 

refracting,  of  prisms,  291. 

sine  of,  286. 

tangent  of,  372. 

visual,  314. 
Anode,  408. 
Antinode,  245. 

Aperture,  of  mirror,  274,  282. 
Arc,  electric,  385. 
Archimedes,  principle  of,  24,  25. 


Armature,   of  dynamos   and    motors, 

398. 
Artesian  wells,  20. 
Astatic  galvanometer,  383. 
Atmosphere,  heating  of,  172-173. 

height  of,  23- 
Atmospheric  electricity,  422-425. 

pressure,  31-36. 

refraction,  294-296. 
Attraction,  electrostatic,  412. 

magnetic,  340. 

molecular,  147. 

of  gravitation,  98-103. 
Audibility,  limits  of,  214,  234-235. 
Aurora  borealis,  425. 
Axis,  of  lens,  297,  300. 

of  mirror,  274. 

Balance,  12-13. 
Balloon,  47. 
Barometer,  34-36. 
Battery,  see  Cells. 
Beam,  of  light,  259. 
Beats,  230-231. 
Bell,  electric,  365. 
Bellows,  43. 
Bichromate  cell,  359. 
Bicycle,  134. 
Boiling,  200-202. 
Boiling  point,  175,  202. 
Boyle's  law,  39-40. 
Buoyancy,  center  of,  65. 

of  air,  46-48.  V 

of  liquids,  24-25. 


431 


432 


Index 


Caloric  theory,  i6l. 
Calorie,  182. 

Calorimetry,  182-185,  190,  205. 
Camera,  photographer's,  322. 

pinhole,  263. 
Capillarity,  153-155- 
Cathode,  408. 
Cells,  electric,  354,  357-362. 

electro-motive  force  of,  376-377. 

in  battery,  380-382. 

storage,  410. 
Center,  of  buoyancy,  65. 

of  curvature,  274. 

of  gravity,  60-61. 
Centrifugal  force,  ^6. 
Centripetal  force,  93-96. 
Charge,  electrostatic,  420. 
Charles,  law  of,  179,  181. 
Chemicar  changes,  i. 

effects  of  electric  current,  407-411. 

energy,  120. 
Chromatic  aberration,  333. 

scale,  234. 
Circuit,  electric,  357. 

divided  or  shunt,  378-379. 
Clouds,  I9'i-I99. 
Coal,  energy  of,  iii,  120,  210. 
Cohesion,  147-151.  * 

Coil,  induction,  393. 
Color.  325,  327-333. 

by  interference,  337. 
Commutator,  397. 

Compass,  340,  348,  352,  ^ 

Compressibility,  of  gases,  4,  37-40. 

of  liquids  and  solids,  4,  140. 
Compression  pump,  43, 
Condensation,  of  gases,  205-206. 

of  water  vapor,  198-200. 
Condenser,  electric,  420. 
Conduction,  electric,  357. 

of  heat,  164-165.. 
Conservation,  of  energy,  132,  209. 
Convection,  of  heat,  166. 
Couple,  58. 
Critical  angle,  293. 


Crookcs  tubes,  422. 

Current,  electric,  354-355.  369-373- 

chemical  effects  of,  407-411. 

extra,  392. 

first  ideas  of,  354-355- 

heating  effects  of,  385-387. 

induced,  389-396. 

magnetic  eflects  of,  362-365. 

measurement  of,  369-373,  384. 

unit  of,  377. 
Curvature,  center  of,  274. 
Curvilinear  motion,  93-^. 

Dahon's  laws,  195. 
Daniell  cell,  362. 
D'Arsonval  galvanometer,  383. 
Declination,  magnetic,  349. 
Density,  13. 

and  pressure  of  gases,  40. 
Deviation,  angle  of,  284,  291. 
Dew,  199. 
Dew-|X)int,  196. 
Diffusion,  of  gases,  141-144. 

of  liquids,  145-146. 

of  light,  269-270. 
Dip,  magnetic,  350. 
Dipping  needle,  350. 
Discord,  232. 

Dispersion,  of  light,  322-326. 
Distillation,  203-204. 
Divisibility,  of  matter,  137-138. 
Ductility,  159. 
Dynamics,  Chap.  V. 

definition  of,  49. 
Dynamo,  396-400. 

Ear,  249-252. 

Earth,  effect  of  rotation  on  its  shape, 

lOI. 

effect  of  rotation  on  weight,  102. 

magnetic  field  of,  348-352. 

revolution  and  rotation  of,  100. 
Echoes,  225. 
Eclipses,  261-262. 
Efficiency,  of  machines,  132. 


Index 


433 


Elasticity,  156-158. 
Electric,  arc,  385. 

battery,  380-382. 

bell,  365. 

cells,  354-355»  357-362. 

circuit,  357. 

conduction,  357. 

current,  see  Current. 

discharge,  in  rarefied  gases,  421. 

energy,  354-355.  387.  389,  392.  399, 
401,  402,  407. 

light,  385-386. 

measurements,  369-384. 

motors,  400-402. 

potential,  see  Potential. 

power,  387. 

resistance,  see  Resistance. 

spark,  395. 

telegraph,  366-369. 

transmission  of  power,  402. 

units,  375,  376,  377,  388. 
Electricity,  Chap.  XII. 

atmospheric,  422-425. 

•current,  353-4". 

nature  of,  353,  413. 

static,  411-425. 
Electrification,  by  contact,  413. 

by  friction,  412. 

by  induction,  413-414. 

two  states  of,  413. 
Electrodes,  408. 
Electrolysis,  407-410. 
Electro-magnet,  364. 
Electro-magnetic  field,  362-364. 

induction,  389-396. 
Electro-metallurgy,  409. 
Electro-motive  force,  356. 

measurement  of,  379. 

of  cells,  376-377- 

uftit  of,  376.  • 

Electrophorus,  416. 
Electroplating,  409. 
Electroscope,  415. 

Electrostatic  attraction  and  repulsion, 
412. 


capacity,  419. 
charge,  420. 
condenser,  420. 
field,  415. 
induction,  413. 
machines,  415-417. 
potential,  418. 
Electrotyping,  408. 
Energy,  Chap.  VI. 

conservation  of,  132,  209. 
dissipation  of,  1 21,  223. 
first  ideas  of,  ii i . 
forms  of,  chemical,  120. 
electrical,  354-35 5»  387.  389.  392, 

399,401,402,407. 
kinetic,  in,  116. 
mechanical,  120. 
molecular    kinetic    (heat),    118, 

146-147. 
molecular      potential       ("latent 

heat"),    189,   204-205. 
muscular,  119. 
of  light,  255-258. 
of  sound,  220-223. 
potential,  118,  1 19,  1 20. 
radiant,  167-172,255-258. 
solar,  209-211. 
sources  of,  162-163,  210. 
transference  of,  by  electric  current, 
354-355»  402. 
by    machines,      123,     126,     128, 

130. 
conditions  necessary  for,  112.    , 
two  modes  of,  255. 
transformation   of,   by   dombustion, 
163. 
by  compression  and  expansion  of 

gases,  162,  204. 
by  dynamos,  399. 
by  electrical  resistance,  387. 
by  friction,  119,  161-162. 
by  fusion  and  solidification,  189. 
by  motors,  401. 

by  radiation  and  absorption,  168, 
170. 


434 


Index 


by  steam  engine,  213. 
by  vaporization  and  condensation, 
193,  204-207. 
units  of,  electrical,  387-388. 
mechanical,  116. 
thermal,  182. 
Engine,  steam,  2 12-2 1 3. 
Equilibrant,  54. 

Equilibrium,  of  concurrent  forces,  52. 
of  floating  bodies,  65-66. 
of  parallel  forces,  55-56. 
of  two  forces,  49-50. 
stable,  unstable,  and   neutral,  60- 

63. 
Ether,    luminiferous,    256-258,    347, 

353. 
Evaporation,  192. 

cooling  by,  206-207. 
Expansion,  by  heat,  141,  176-180. 

cooling  by,  204. 

force  of,  177. 
Extension,  10. 
Eye,  306-315. 

defects  of,  310. 

Falling  bodies,  74-79. 

Faraday,  392. 

Far  sight,  310. 

Field,  electro-magnetic,  362-364. 

electrostatic,  415. 

magnetic,  345-347- 
Field  magnet,  399. 
Floating  bodies,  buoyancy  upon,  24- 

25. 

equilibrium  of,  65-66. 
Fluids,  characteristics  of,  4-5. 
Focal  length,  of  lens,  299. 

of  mirrors,  276,  282. 
Foci,  of  lenses,  297-300. 

of  mirrors,  276,  279. 
Fog,  198. 
Foot-pound,  116. 
Force,  5-9. 

buoyant,  24,  25,  46-48. 

centrifugal,  96. 


centripetal,  93-96. 

electro-motive,  356. 

elements  of,  50. 

graphic  representation  of,  50. 

lines  of,  345-347- 

moments  of,  57-59. 

units  of,  12. 
Forces,  balanced,  7,  8;  Chap.  IV. 

composition  of,  51-53,56. 

concurrent,  51-55. 

molecular,  147-155. 

parallel,  55-56. 

parallelogram  of,  52. 

resolution  of,  54. 

unbalanced,  7,  8;  Chap.  V. 
Force  pump,  45. 
Franklin,  422,  423, 425. 
Fraunhofer  lines,  324. 
Freezing,  186. 
Freezing  mixtures,  191. 
Freezing  point,  1 74. 
Friction,  6. 

heating  effects  of,  119,  161-162. 

of  the  air,  74,  75,  78. 

uses  of,  90,  133,  134. 
Frost,  199. 
Fulcrum,  123. 

Fundamental  tone,  238,  246-247. 
Fusion,  186-191. 

change  of  volume  during,  187. 

heat  of,  189-190. 

Galilean  telescope,  320. 
Galileo,  76,  79,  92,  320. 
Galvanometers,  370-373,  383-384. 
Gases,  characteristics  of,  3. 

compressibility  of,  4,  39,  40. 

condensation  of,  205-206. 

cooled  by  expansion,  204. 

diffusion  of,  141-143. 

distinguished  from  vapors,  192. 

effect   of  pressure  on  volume  and 
density  of,  39, 40.  • 

effect  of  temperature  on  volume,  of, 
141,  179. 


Index 


435 


kinetic  theory  of  144-145. 

mechanics  of,  Chap.  III. 

pressure  of,  37-40. 
Geissler  tubes,  421. 
Glaciers,  flow  of,  189. 
Gram,  mass  and  weight,  12. 
Gram-centimeter,  116. 
Gravitation,  98-103,  150-151. 

law  of,  98. 
Gravity,  60. 

acceleration  due  to,  76-78. 

cell,  360. 

center  of,  60,  61. 

specific,  26-30. 

Hail,  200. 
Hardness,  160. 
Harmony,  232. 
Hearing,  249-252. 
Heat,  Chap.  VHI. 

conduction  of,  164-166. 

convection  of,  166. 

expansion  due  to,  141,  176-180. 

kinetic  theory  of,  146-147,  161-162. 

mechanical  equivalent  of,  208. 

of  fusion,  189-190. 

of  vaporization,  193,  204-205. 

sources  of,  162. 

specific,  183-184. 

unit  of,  182. 
Helmholtz,  211,  240. 
Horse  power,  121. 
Humidity,  197. 
Hydrostatic  press,  21. 

Ice,  174,  186-190. 

manufacture  of,  207. 
Illumination,  intensity  of,  263. 
Images,  by  lens,  300-304. 

by  plane  mirrors,  271-273. 

by  spherical  mirrors,  280-283. 

by  small  opening,  262-263. 

real,  279-281,  300,  302.  • 

virtual,  279-283,  302,  303. 
Incandescent  lamp,  385. 


Inclination  or  dip,  350. 

Inclined  plane,  54,  79,  1 14,  131-132. 

Index  of  refraction,  287. 

Induced  currents,  389-396. 

Induction,  earth's,  351-352. 

electro-magnetic,  389-396. 

electrostatic,  413. 

magnetic,  342-343. 

self,  392. 
Induction  coil,  393. 
Inertia,  5,  87,  89,  95,  96. 
Insulators,  electric,  357. 
Interference,  of  light,  337. 

of  sound,  229-231. 
Iridescence,  337,  338. 

Joule,  162,  208,  386. 
Joule's  equivalent,  208. 
law,  386. 

Kilogram-meter,  116. 
Kinetic  energy,  III,  Il6. 
Kinetic  theory,  of  gases,    144,    146- 
147. 
of  heat,  146-147,  161-162. 
Kinetics,  see  Dynamics. 

Lamp,  arc,  385. 

incandescent,  385. 
Lantern,  optical,  321. 
Law,  Boyle's,  39-40. 

Dalton's,  195. 

Joule's,  386. 

»f  Charles,  179,  181. 

of  gravitation,  98. 

Ohm's,  370,  377. 

Pascal's,  20. 
Laws  of  motion,  Newton's,  91. 
Leclanche  cell,  360. 
Lenses,  achromatic,  334. 

concave,  305. 

convex,  296-305.  % 

Lever,  123-126. 
Leyden  jar,  418. 
Lifting  pump,  44. 


436 


Index 


Light,  Chap.  X. 

dispersion  of,  322-326. 

intensity  of,  263-265. 

interference  of,  337.* 

propagation  of,  258-260. 

reflection  of,  268-270. 

refraction  of,  283-292. 

theory  of,  255-260. 

velocity  of,  265-267. 

wave  length  of,  259,  325-326. 
Lightning,  422-423. 

rod,  423. 
Lines  of  force,  345-347- 
Liquids,  characteristics  of,  3,  5. 

diffusion  of,  145-146. 

mechanics  of,  Chap.  IL 
Liter,  11. 

Local  action  in  voltaic  cell,  358. 
Lodestone,  339. 
Loudness  of  sound,  227. 

Machines,  123-135. 

efficiency  of,  132. 

electrical,  415-417. 

mechanical  advantage  of,  126. 
Magdeburg  hemispheres,  32. 
Magnetic  action,  340-341,  347. 

declination,  349. 

effects  of  a  current,  362-365. 

field,  345-347- 

inclination  or  dip,  350. 

induction,  342-345. 

lines  of  force,  345-347. 

meridian,  349. 

needle,  340. 

permeability,  342. 

poles,  340. 

substances,  341. 
Magnetism,  Chap.  XL 

terrestrial,  348-352. 
Magnetization,   permanent   and 
porary,  342-343- 

theory  of,  343-345- 
Magnets,  338,  342. 
Magnifying  glass,  302,  315-316. 


tem- 


Major  chord,  234. 
Malleability,  159. 
Manometers,  38-39. 
Mass,  detinition  of,  11. 

center  of,  60-61. 

measurement  of,  by  weight,  12, 
by  inertia,  87. 

units  of,  12. 
Matter,  divisibility  df,  137. 

properties  of.  Chap.  VI L 

states  of,  3,  147-148. 

structure  of,  137-147. 
Measurement,  10-13. 
Mechanical  advantage,  126. 

equivalent  of  heat,  208. 

powers,  123. 
Mechanics,  definition  of,  49. 

of  gases.  Chap.  IIL 

of  liquids.  Chap.  IL 

of  solids.  Chaps.  IV,  V,  VI. 
Melting,  186. 
[Celling  points,  186. 
'    effect  of  pressure  on,  1 88. 
Meter,  lo. 

Microphone,  405-406. 
Microscope,  compound,  316. 

simple,  302,  315-316. 
Mirage,  295. 
Mirrors,  parabolic,  282. 

plane,  271-273. 

spherical,  274-283. 
Mobility  of  fluids,  159.  • 
Molecular  forces,  14 7-1 51. 

motion,  141-147. 

structure  of  matter,  137-147. 
Molecule,  138. 
Moment  of  force,  57-59. 
Momentum,  91. 
Moon,  revolution  of,  100. 
Motion,  68. 

accelerated,  69,  71-74 

curvilinear,  93-96. 

laws  of, '83-92. 

of  falling  bodies,  74-78. 

of  pendulum,  104. 


Index 


437 


of  projectiles,  80-83. 

on  an  inclined  plane,  79. 

uniform,  68. 

wiave,  217-220. 
Motor,  electric,  400-402. 
Musical,  instruments,  242,  246-248. 

intervals,  231-233. 

scales,  233-234. 

sounds,  227. 

Needle,  dipping,  350. 

magnetic,  340. 
Newton,  92,  98,  255,  259. 
Newton's  disks,  331. 

laws  of  motion,  91. 
Nodes,  237,  244-245. 
Noise,  227,  245. 

Octave,  232. 

Ohm,  definition  of,  375. 

Ohm's  law,  370,  377. 

Opera  glass,  321. 

Optical  instruments,  315-322. 

Organ  pipes,  246-247. 

Overtones,  238,  246-247. 

Parallelogram  of  forces,  52. 

Pascal's  law,  20. 

Pendulum,  104-109. 

Penumbra,  261. 

Permeability,  magnetic,  342. 

Photometry,  264-265. 

Physical  changes,  i. 

Pinhole. camera,  263. 

Pitch,  of  musical  sounds,  214,   227- 

229,  234. 
Plasticity,  149,  158. 
Polarization,  in  voltaic  cell,  358. 
Poles,  magnetic,  340. 

of  voltaic  cell,  355,  358-359- 
Porosity,  138-139. 
Potential,  electric,  356,  377. 

of  induced  currents,  395. 

of  static  electricity,  418. 
Potential  energy,  118,  119,  120. 


Pound,  weight  and  mass,  12. 
Power,  121. 

electric  transmission  of,  402. 

units  of,  121,  387. 
Pressure,  atmospheric,  31-36 

of  gases,  37-40. 

of  liquids,  15-25. 

of  vapors,  193-195. 
Pressure  gauges,  33-35,  38-39- 
Prism,  291. 
Projectiles,  80-83. 
Properties  of  matter.  Chap.  VII. 
Pulleys,  129-130. 
Pump,  air,  41. 

compression,  43. 

force,  45. 

suction,  44. 

Quality  of  sound,  227,  238-240. 

Radiant  energy,  167-172,  255-258. 
absorption  of,  168-169,  171,  172. 
emission  of,  168. 
luminous   and    non-luminous,    168, 

257. 

reflection  of,  170,  268. 

selective  absorption  of,  171,327-329. 

transmission  of,  169,  328-329. 
Radiometer,  170. 
Rain,  199. 
Rainl>ow,  335-336- 
Ray,  of  light,  259. 
Reaction  and  action,  8,  9,  89,  90,  340, 

413. 
Reflection,  of  light,  268-283. 

difi'used,  269-270. 

of  sound,  225. 

regular,  268-269,  291. 

total,  292-294. 
Refraction,  283-292. 

atmospheric,  294-296. 

index  of,  287. 

laws  of,  286. 

of  different  colors,  322-326. 

relation  to  velocity,  28S-289. 


438 


Index 


Relay,  368. 

Resistance,  electrical,  357. 

laws  of,  374. 

measurement  of,  376. 

of  conductors  in  parallel,  378. 

specific,  374. 

unit  of,  375. 
Resistance  coils,  375. 
Resolution,  of  a  force,  54. 

of  a  velocity,  70. 
Resonance,  243-245. 
Respiration,  46. 
Resultant  force,  54. 

velocity,  70. 
Retina,  307. 
Rumford,  Count,  l6l,  265. 

Scales,  musical,  233-234. 

Screw,  133. 

Selective  absorption,  171,  327-329. 

Self-induction,  392. 

Shadows,  260. 

Short  sight,  310. 

Shunt  circuit,  378-379. 

Sine  of  an  angle,  286. 

Siphon,  45. 

Size,  angular,  314. 

Sky,  the  color  of,  330. 

Snow,  200. 

Soap  bubbles,  152. 

Solenoid,  364. 

Solidification,  186-189. 

Solids,  characteristics  of,  3,  5. 

mechanics  of.  Chaps.  IV,  V,  VI. 
Solution,  heat  of,  191. 
Sonometer,  232. 
Sound,  Chap.  IX. 

intensity  of,  220-223. 

interference  of,  229-231. 

loudness  of,  227. 

media,  216. 

pitch  of,  214,  227-229,  234. 

properties  of,  227. 

quality  of,  227,  238-240. 

reflection  of,  225. 


sources  of,  214. 

velocity  of,  224-225. 

waves,  218-220,  229. 
Sounder,  telegraph,  366. 
Speaking  tubes,  223. 
Specific  gravity,  26-29. 

heat,  183-184. 

resistance,  374. 
Spectrum,  322-326. 

invisible,  326. 
Speed,  68. 

Spherical  aberration,  282,  304. 
Spyglass,  320. 
Stability,  64. 

of  floating  bodies,  66. 
Stars,  distance  of,  267. 

twinkling  of,  294. 
Static  electricity,  411-425;   see  Elec- 
trostatic. 
Statics,  of  solids,  Chap.  IV. 
Steam  engine,  212-213. 
Stereoscope,  313. 
Stress  and  strain,  158. 
Strings,  vibration  of,  235-240. 
Sun,  energy  of,  209-211. 
Surface  tension,  151-153. 
Sympathetic  vibrations,  24^^248. 

Tangent  galvanometer,  370-373. 
Tangent  of  an  angle,  372. 
Telegraph,  365-369- 
Telephone,  404-407. 

acoustic,  223. 
Telescopes,  317-321. 
Temperature,  163-164. 

absolute,  180. 

measurement  of,  173-176. 
Tenacity,  148. 

Terrestrial  magnetism,  348-352. 
Theory,  definition  of,  145. 

of  electricity,  353,413- 

of  gases,  144-145,  146-147. 

of  heat,  146,  1 61-162, 

of  light,  255-260. 

of  magnetic  action,  347. 


Index 


439 


of  magnetization,  343-345. 

of  the  structure  of  matter,  137-141. 
Thermometers,  173-176. 
Thunder,  423. 
Tone,  233. 

fundamental,  238,  246. 
Torricelli,  35. 
Total  reflection,  292-294. 
Transference  of  energy,  see  Energy. 
Transformation  of  energy,  see  Energy. 
Transformer,  403. 
Tuning  fork,  pitch  of,  228. 

vibration  of,  215. 

Umbra,  261. 

Units,  of  acceleration,  72. 

of  current  strength,  377. 

of  electrical  power,  387. 

of  electrical  resistance,  375. 

of  E.  M.  F.,  376. 

of  extension,  lo-ll. 

of  fluid  pressure,  39. 

of  force,  12. 

of  heat,  182. 

of  mass,  1 2. 

of  mechanical  power,  1 21. 

of  velocity,  68,  69. 

of  work  and  energy,  116. 

Vacuum,  35. 

Vapor,  atmospheric,  196-200. 

pressure  of,  193-195. 
Vaporization,  192,  198. 

heat  of,  193,  204-205. 
Vapors,  192. 
Velocity,  68. 

graphic  representation  of,  69. 

of  light,  265-267. 

of  sound,  224-225,  229. 

resolution  of,  70. 


uniform,  68. 

variable,  69-74. 
Velocities,  composition  of,  69. 
Vibration,    forced    and     sympathetic, 
240-242. 

of  air  columns,  243-248. 

of  molecules,  146. 

of  pendulum,  104-105. 

of  strings,  235-240. 

of  tuning  fork,  215. 
Viscosity,  159. 

surface,  153. 
Vision,  binocular,  311-314. 
Visual  angle,  314. 
Vocal  cords,  253. 
Voice,  252-254. 
Volt,  definition  of,  376. 
Voltaic  cell,  354. 
Voltmeter,  379,  384, 

Water,  expansion  of,  178,  179. 
Watt,  388. 
Wave  motion,  217. 

Waves,   of  light,  257,  259-260,  325- 
326. 

of  sound,  218-220,  229. 

of  water,  218. 
Weather,  prediction  of,  36. 
Wedge,  133. 
Weighing,  12. 
Weight,  7,  II,  102. 
Welding,  149. 
Wheel  and  axle,  127-128. 
Whistle,  245. 

Wind  instruments,  246-248. 
Windlass,  128. 
Work  and  energy,  311-316. 

units  of,  116. 

Zero,  absolute,  1 80-1 81. 


THIS  BOOK  IS  DUE  uN  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVEROUe. 


SEP  13  1934 


ACT  19  IS74  17 


s^ 


¥^ 


0^ 


^ 


I,n  21-10O,;i-7.'33 


VB 


^5Ar 


^D4u 


^ 


^ 


^    6/-" 
^^« 


UNIVERSITY  OF  CALIFORNIA  UBRARY 


